Let me give you a kind of off-the-wall answer and maybe it helps. To do this I am going to drop down one dimension, from volume to area, and I am going to give you kind of what calculus “is” in my way of looking at it.
A curiosity about circles
So when I was a kid, I was forced to memorize that we define $\pi$ as the ratio of a circle’s circumference to its diameter. (It was not absolutely obvious to me at that time that this should be a fixed numerical ratio. This is also a problem that can be solved with calculus-thinking. But I just had to take it on adults’ authority that somehow all lengths, even curved ones, in a figure must scale with the scaling parameter. So when you zoom in by a factor of 2 the diameter and circumference both double and the ratio remains fixed. The home that calculus is needed to prove it furnishes some nice counterexamples in fractals.) So I could accept that we called it $\pi$ and it was measured as approximately 3.14159.
But if that was maybe mildly surprising, far more outrageous was it that the area of the circle was $\pi r^2.$ Like, I memorized this very quickly. It is very memorable. But, why is it the same $\pi$? Why not, say, $\pi^2$? Okay, well, maybe not $\pi^2,$ because $\pi^2$ is obviously too large. I was able to see that the circle was inscribed within a square of side-length $2 r$ and therefore it must be less than $4.$ With some cleverness I could inscribe a dodecagon to find that $A>3r^2$, furthermore I could stretch the dodecacon out to find after a lot of work that $A< 6 (\sqrt{3} - 1)^2,$ so it had to be less than 3.2154 and if I guessed halfway between those I would get 3.11 or so. This was already enough to exclude, say, $\pi^2/3$.
But still, that question of “how can I see that this is exactly $\pi$” was missing for me until I learned calculus, and I learned that it has two distinct proofs in calculus, one which we call “integral” calculus and one which we call “differential calculus.” Curiously, they have both to do with Italy’s most famous foods.
Proof by pizza
Slice the circle with a perfect blade into $N$ slices, pizza-like. Then rearrange them, stacking $N/2$ of those slices pointed “up” with the other $N/2$ pointed down, to “sicilianize” the pizza into a sort of almost-parallelogram shape. As $N$ gets very very large we would be creating these infinitesimally thin almost-triangles of pizza! (There is your counterexample, infinitesimal triangles instead of infinitesimal squares.) And the “crust” of the pizza needs to be distributed exactly over the top and the bottom of the parallelogram whereas the larger $N$ gets the more this needs to look like a rectangle. So we have a rectangle of side length $\pi r$ and height $r$ and so it must have area $\pi r^2.$ Proof by pizza.
You actually don't have to rearrange them, just the fact that there are $N$ triangles with areas $\frac12~r~(\2\pi r/N)$ already gives you $\pi r^2$ when you add them all together. The “Sicilianize them” step is just a nice touch that I am stealing from a quantum complexity theorist named Scott Aaronson, heh.
This we call the “integral calculus”, it is about trying to cut up a complex shape into a lot of tiny “infinitesimal” simple shapes, and then rearrange them or sum them back up. In this case the shapes are simple because they are, in the limit of large $N$, triangles. The core idea of calculus is that when I zoom in on this circle’s edge enough, the circle looks like a straight line, so if I take very small chunks of it I can pretend they are not pizza slices but triangles.
Proof by pasta
Here's a very different sort of proof that calculus also offers. It says that I can make a circle which is slightly larger, by wrapping a thin piece of spaghetti around an existing circle. This thin piece of spaghetti can then be unwrapped: it has length $2\pi r,$ roughly, and width $\delta r$. Meanwhile since we know the area scales quadratically with our zoom factor, we know the area is $A = \alpha r^2$ for some $\alpha$, and this says that $$\alpha (r + \delta r)^2 \approx \alpha r^2 + 2 \pi r~\delta r.$$ Ignoring the $\delta r^2$ term (which is a tiny little triangle chunk of spaghetti at the end, the spaghetti was actually a sort of trapezoid with one edge being $2\pi(r + \delta r)$ and the other being only $2\pi r$: ignore the little chunk), we FOIL out the product on the left and find out that $$\alpha r^2 + 2 \alpha r~\delta r \approx \alpha r^2 + 2 \pi r~\delta r$$ and we conclude that $\alpha$ must have been $\pi$ all along.
More esoteric proof constructions
You can also use both of these the other way!
You can do the integral proof with spaghetti: create the circle as $N$ nested circles of spaghetti of width $r/N$. The length varies, but the $k^\text{th}$ one corresponds to the circle of radius $k r/N$, so when we unroll all of these we get a sort of rough triangle with height $r$ and base $2\pi r$ and so it must have area $\pi r^2$ after we compute the triangle’s $A = \frac12 b h.$
Or the differential proof with pizza: cut a radius in the circle and try to stretch it open a little bit, an opening $\delta C$ in terms of circumference length. We want to say that we can reshape that dough into a slightly larger circle with the same area, so the old area was $\alpha r^2$, this has been redistributed into a chunk of a circle of new radius $r + \delta r$, that chunk we can measure as being $(C - \delta C)/C$ of the new circle. So the new radius must be given by
$$ \alpha (r + \delta r)^2 \left(1 - \frac{\delta C}{C}\right) \approx \alpha r^2,\\
\delta r \approx \frac{\delta C}{4\pi}.\\
$$
Once you have this you can finish the argument that $\alpha (r + \delta r)^2 \approx \alpha r^2 + \frac12 r~\delta C = \alpha r^2 + 2\pi r~\delta r,$ because we add the missing pizza slice with area $\frac12 r~\delta C.$ It's a weirder argument but you can certainly make it.
What this says about infinitesimals
I said above that the key point about calculus is that when you zoom way way in on a circle it looks like a straight line, and we have now added an infinitesimal pizza slice to an existing circle, and we have cut it into infinitesimal pizza slices so we could rearrange those slices into a breadtangle of pizza: both of these are based on this insight that the pizza slices become like triangles. But we also see something similar with the spaghetti: we are either adding an infinitesimal bit of spaghetti around the edge of a circle and then unrolling it, or else we are building the whole think out of concentric circles of spaghetti: but what we have in common is that because locally the circle looks like a straight line, the noodles become floppy and can be easily unrolled into being flat.
Now, infinitesimals are this helpful mental tool for a way of talking about this sort of argument. The claim is that if the spaghetti is thin enough then who cares about a tiny little chunk off the end of the spaghetti; if the pizza slice is thin enough then who cares about a tiny little curve of its crust?
One way to make this rigorous is to think, “If I made the spaghetti half as thin, then the little chunk on the end would occupy only a quarter of the area compared to the spaghetti itself having half the area, so this argument that ignores this chunk gets twice as precise. So I can do this halving however many times I need to do, in order to make this argument as close to correct as I need to.” This is roughly what the definition of limits gives you. It does not define “infinitesimal” directly, it just says that the “infinitesimal” argument is the “limit” of macroscopic arguments and is arguing about certain terms disappearing faster than other ones.
You have also by now seen the hyper-real numbers of “nonstandard analysis.” This is a different mental toolkit to make the same thinking rigorous. In this mental toolkit we “imagine that there are numbers which are so big that you are never going to run into them, you don’t even have the matter in the universe to write them down with some of your finest of chained-arrow notations: super-large numbers. I am not even going to tell you what $N$ is but just to say that after some unspecified number $N$ the numbers become too large for us to care about. Surely this should happen eventually, as numbers become so big that we can’t compute them or think about them.” So that is how we start to formalize the new number system. We can then also have a category of numbers formed by 1/(super-large) that are super-small. These are how we think of infinitesimals.
Part of the hyper-reals is that there’s always half of an infinitesimal, and half of that: just like there's always twice of a super-large number and twice that. And we can fudge $N$ to say that most of these numbers are not near $N$ so that within some bounds of reason there is always twice a super-small number, and half of a super-large one (we just assume that it's way way larger than $N$), as long as we don’t start doing some very suspicious things with them like many-repeated divisions. So we really just invent a number system which has infinitesimal numbers and then we use infinitesimals directly, rather than thinking about how certain expressions with our ordinary numbers limit to various other expressions as we make our arguments smaller and smaller.
There are probably other ways to make this reasoning rigorous, too. But all of that is about justifying these arguments about zooming way in on a problem and approximating the solution with simpler shapes. They don’t have to be squares, they may be triangles or spaghetti.
Coming back to your question
So now you’re me and you’re asked this strange question about whether the infinitesimal square is the smallest infinitesimal area. And the answer is, this misses all of the squishiness of infinitesimals. Like, the category of question is wrong because it assumes $\mathrm dx~\mathrm dy$ is somehow some real objective thing, $\mathrm dx$ being the smallest possible increment in $x$ or so. But the arguments above are all about saying that when I am looking at some finite $\delta x$, I can always look at $\delta x/100$ to get a better approximation. If I am looking at some square $\delta x~\delta y$, I can always cut it diagonally into two triangles if that is preferable. If I am cutting into a million slices of pizza and summing them up and somehow that’s not right and my pizza crust is still too curved, then I will cut into a billion slices of pizza. Or a googol. Or a googolplex. Or Graham's number. Or something that puts Graham's number in the dust.
If I wanted to do discrete calculus, I can also do that, by the way. There is a discrete calculus where we have infinite sequences $x_0, x_1, \dots$ and we define operators like $$(\Delta x)_i = \begin{cases}x_0,& \text{if } i = 0,\\
x_i - x_{i-1}, & \text{otherwise}\end{cases},\\
(\Sigma x)_i = \begin{cases} x_0,& \text{if } i = 0,\\
x_i + (\Sigma x)_{i - 1}, &\text{otherwise}\end{cases}.$$ So for example we can start from the sequence $N = [1, 2, 3, 4, \dots ]$ and form the sequence of odd numbers $N_\text{odd}= 2N - 1 = [1, 3, 5, 7, \dots ]$ and then we can perform $\Delta N_\text{odd} = [1, 2, 2, 2, \dots].$ Or we can perform $\Sigma N_\text{odd} = [1, 4, 9, 16, ...] = N^2.$ There is this discrete calculus with a fundamental theorem that plus undoes minus and minus undoes plus: $$\Delta \Sigma x = \Sigma \Delta x = x.$$
And then we do have your idea of a smallest possible increment, because sequences $x_n$ are like functions $x(n)$ where the smallest possible increment is this rigid $\mathrm dn = 1$. Similarly I have seen $\mathrm dn$ be a rigid “one bit-flip” in the exciting field of differential cryptography which fundamentally changed a lot of how we design security primitives.
So like there exist these other interesting fields. But in this context of normal real analysis, the key thing about differentials is that they are squishy and I can consider long skinny differentials that I wrap around shapes or skinny pizza differentials that I shove into existing pizzas. That squishiness is why I love them. The idea of a rigid unsquishy $\mathrm dx ~\mathrm dy$ underneath that is just unappealing to me.
