I'm in doubt whether the equal sign in an expression like "$0 ^\circ \mathrm{C} = 273.15 \,\mathrm{K}$" is fair, because normally, if $A=B$, then, say, $2A=2B$, which is not applicable to "$0 ^\circ \mathrm{C} = 273.15 \,\mathrm{K}$".
Is it okay to use "$=$" if we cannot actually perform multiplication or division by the same number on both sides? If "$=$" doesn't work, then what symbol is more suitable?
I'm in doubt whether the equal sign in an expression like "$0 {\ ^\circ\mathrm{C}} = 273.15 {\ \mathrm{K}}$" is fair because, normally, if $A = B$, then, say, $2 A = 2 B$, which is hardly applicable to "$0 {\ ^\circ\mathrm{C}} = 273.15 {\ \mathrm{K}}$".
I think that the equals sign is totally appropriate, and the rule "if $A = B$, then $2 A = 2 B$" is perfectly applicable to this equation.
The only tricky part is that, unlike most unit symbols, the unit symbol $^\circ\mathrm{C}$ can't be treated as though it were a quantity that is multiplied by the number next to it. It must be treated as an operator which is written on the right-hand side of the quantity that it operates on.
As a result of this, the rule for multiplying a temperature in degrees Celsius by a scalar is strange. Specifically, the rule is that
$$n (t {\ ^\circ\mathrm{C}}) = (n t + (n - 1) 273.15) {\ ^\circ\mathrm{C}}.$$
But in any case, by applying this rule, we can see that
$$ \begin{align} 2 (0 {\ ^\circ\mathrm{C}}) &= (2 \cdot 0 + (2 - 1) 273.15) {\ ^\circ\mathrm{C}}\\ &= (2 \cdot 0 + 1 \cdot 273.15) {\ ^\circ\mathrm{C}}\\ &= (0 + 273.15) {\ ^\circ\mathrm{C}}\\ &= 273.15 {\ ^\circ\mathrm{C}}\\ &= (273.15 + 273.15) {\ \mathrm{K}}\\ &= (2 \cdot 273.15) {\ \mathrm{K}}\\ &= 2 (273.15 {\ \mathrm{K}}).\\ \end{align} $$
Let the rod's temperature be $x$ kelvins and $y$ degrees Celsius and $z$ degrees Fahrenheit.
The Kelvin and Celsius and Fahrenheit scales don't have coincident zeroes.
As a matter of fact, whereas $$\text{metre}=39.37\times\text{inch},$$ there is no algebraic relationship between the units K and °C and °F: $$100\text{°C}=373.15\text{ K} \kern.6em\not\kern-.6em\implies\\ \text{°C}=\text{K}\pm273.15\quad\text{or}\quad\text{°C}=3.7315\text{ K}.$$
Therefore, temperature conversion is not generally via equality but certainly a bijection (one-to-one correspondence): $$\boxed{\;x\text{ K} \ne y\text{°C} \ne z\text{°F}\,.\\{}\\ \;x\text{ K} ≅ y\text{°C} ≅ z\text{°F}\quad\iff\\x=y+273.15\;\text{ and }\;y=\frac59(z-32).}$$
On the other hand, the temperature conversion $$\boxed{x\text{ K}=1.8x\text{°R}}$$ is a true equality; in other words, corresponding numeric values of degrees Rankine and kelvins are in proportion (as are the units themselves: $\text{K}=1.8\text{°R}).$
Incidentally, among the temperature states $1.8x\text{°R},$ $x\text{ K},$ $y\text{°C}$ and $z\text{°F},$ only the first two are actually physical quantities, since only they lie on ratio scales, since only they have true zeroes (and genuine magnitudes).
Is the issue that the mathematical zero-level has to correspond to a physical zero-level, or is the issue simply that the two mathematical zero-levels are not coinciding?
The latter (this issue is purely mathematical): we can for instance construct a scale called ‘Pelsius’ such that $t\text{°P}$ actually equals $3t\text{°C},$ with neither scale having a true zero (i.e., neither having a zero point equalling absolute zero).
Finally, all these are false equalities:
Notice that multiplying temperatures isn't generally physically meaningful; for example, consider $3\times(-100)\text{°C}.$
Something like $\text“3\times5\text{°C}=15 \text{°C”}$ probably means that the author is multiplying a temperature interval (temperature difference) rather than a temperature, in which case it is heat energy, rather than displacement from water's freezing point, that has tripled.
Addendum
Wikipedia agrees: “Temperatures on scales that either do not share a numeric zero or are nonlinearly related cannot correctly be mathematically equated (related using the symbol =), and thus temperatures on different scales are more correctly described as corresponding (related using the symbol ≘).”
(Here, the relation $f(x)=mx+c$ is considered merely affine rather than linear.)
EDIT: Apparently the SI convention is to use °C to refer to temperature difference from 273.15K, which would make the OP correct and my previously written answer (below) incorrect: you can't use an equals sign at all, unless both sides of the equality are $T- 273.15K$.
Degrees Celsius is an operator, not a variable. It goes on the right hand side of what it operates on, against mathematical convention for single symbol operators, because of tradition in common use, but that's what it is. It means "take this number and add it to the zero point of the Celsius scale". If we call the zero point $x$ then the operator C acts on the variable $y$ to make $y+x$. If you are precise in treating the degrees Celsius symbol as an operator that is positioned unconventionally, the equation and all algebraic manipulations hold. If you are imprecise and treat the degrees Celsius symbol as a function or a variable, the equation is neither true nor false, it merely inherits the nonsense in the expression, exactly as if you interpreted 2+2 as 2×(+2) and inferred 6×(+2)=12 from 2+2=4.
This is comparing apples & oranges in general.
Example 0 : Water & Tumbler
A tumbler having 50% water is half-full & half-empty :
0.5 full = 0.5 empty.
Multiply each side by 2 to get :
1.0 full = 1.0 empty !
What is wrong here is that we equated the water Part with the empty Part. Naturally, we will get contradictory answers on further manipulations.
This is comparing apples & oranges.
Example 1 : Age & Date
Some Person who was born in 2000 will be 22 years in 2022 AD , But we can not claim that this Person will 2x22=44 years old in 2x2022=4044 AD.
We can not assign a meaning to multiplying Age & Date , & then comparing with each other.
This is comparing apples & oranges.
We can say Age(2022)=22 , 2xAge(2022)=2x22=44 , Age(4044)=2044 or in general Age(X)=(X-2000) which is not Age(2X)=2(X-2000).
Example 2 : Celsius & Kelvin
Absolute 0 is a Point in the Temperature Curve. Multiply (& Dividing) is simply scaling this curve. With a Particular scaling, we Define it Kelvin. Let X=20 Kelvin be a Point on the Kelvin Curve. Multiply by 10 to get 200 Deci-Kelvin.
We have a meaning to use here: 20K=200 Deci-K.
Add (or Subtract) some Constant on the Temperature Curve & we have some other Curve (maybe shifting) and we can not multiply this & then compare with Original Temperature Curve.
This is comparing apples & oranges.
We can , of course, give a new name for this new Curve, eg Celsius.
We can say C=K(X) , where C is Celsius & K is a function converting some temperature in Kelvin to Equivalent temperature in Celsius.
This gives 2C=2K(X) , not 2C=K(2X)
Example 3 : Length & Area
With Side=3 , Area of Square = 9.
We can not claim that with Side=2x3 , Area=2x9=18.
This is comparing apples & oranges.
We can say A=f(S) where A is the Area & f converts length S into Area of Square.
This means, 2A=2f(S), which is true. We can not say 2A=f(2S) which is not true.
Summary :
0C=273.15K actually means that there is a function which converts K to C.
Check Page 3 Equation 3 of this Article which gives the Precise formula.
C=f(K)
0=f(273.13)
Multiply to 2 to get:
2C=2f(K)
0=2f(273.13)
We can not conclude this:
2C=f(2K)
0=f(2x273.13)
This is comparing apples & oranges in general.
When can we use "=" :
(1) When we are not going to Mathematically manipulate it further, Eg by squaring or using trigonometric functions.
[[ In case we want to Mathematically manipulate, we must ensure Mathematical Equality by using the conversion formula ]]
(2) When we are using Differences in temperature , we can multiply without contradictions. Difference in temperature is 2 Units. Double it to get Difference of 8 Units. This is valid when we are using linear scale.
What else can we use to avoid Issues :
(1) We could use words like "Equivalent" & "is same as"
(2) We could also use symbols like $\equiv$ & $\leftrightarrow$
The equal sign is fair, the multiplication is not.
$^\circ C$ does not have a meaningful meaning of multiplication, for any value. One requirement for multiplication to be meaningful, you basically need a zero that has a philosophical meaning of zero (i.e., that there is no quantity of something). The Kelvin scale satisfies this criteria (at $0K$ there is no physical movement), but the Celsius scale does not ($0^\circ C$ is an arbitrary point).
Now, you can multiply changes in $^\circ C$, because changes by definition have a philosophical zero ("no change" actually is in fact zero). But, interestingly, changes in $^\circ C$ is equivalent to changes in $K$.
