It seems to me that the energy of some kind frequently takes the form
$$\frac{a_1a_2^2}{2}$$
Where $a_1$ and $a_2$ are some variables.
So, for example, kinetic energy has $m$ and $v$; rotational energy has $I$ and $\omega$; energy stored inside a spring has $k$ and $x$; capacitor's energy has $C$ and $V$; inductor's enrgy has $L$ and $I$.
Is there any fundamental reasons seemingly many physical phenomenon has the energy in that form?
First, it's not true that energies are generally in the form $\frac{a_1a_2^2}{2}$. Take the gravitational potential energy $U = \frac{GMm}{r}$ as an example. However, it is generally true that kinetic energy takes that quadratic form. Why?
Kinetic energy is the energy traded when some agent applies a force on some system that causes it (the system) to move, that is, to change its state in some fashion as time goes by. The agent is said to do work on the system and, so, the kinetic energy gained by the system is the work done by the force applied by the agent.
Note that I was very vague in the paragraph above about the nature of the force or of the motion of the system. I did that on purpose because I didn't want to refer exclusively to motion in space. There are other ways in which the state of a system could change with time. It doesn't have to be just a change in position.
Now, back to the definition of work. The gain in kinetic energy is given by
gain in KE = $\int_A^B \vec{F}(\vec{s}\,)\cdot d\vec{s}$
where $A$ and $B$ represent the initial and final states of the system, $\vec{F}$ is the force being applied, and $d\vec{s}$ is an infinitesimal displacement of the system. Again, $d\vec{s}$ does not need to be a displacement in position. You could have a meaningful definition of displacement for other properties of a system. For example, displacement (changes) in electric charge.
Likewise, you can have a meaningful definition of velocity that isn't related to motion in space. Velocity is really just the rate of change of something, most commonly position, but you could define other forms of velocity. So, the rate of change of the system's state displacement (for lack of a better term), the velocity associated with a displacement $d\vec{s}$, is just what you'd expect, $\vec{v} = d\vec{s}/dt$. An example in electromagnetism would be the electric current, the rate of change of accumulated charge, $i = \frac{dq}{dt}$.
Now let's try finding the gain in KE not in terms of displacement but in terms of time, by using $d\vec{s} = \vec{v}\,dt$. Then,
gain in KE = $\int_A^B \vec{F}(\vec{s}\,)\cdot\vec{v}\,dt$
Next, recall that $\vec{F}$ is the rate of change in the momentum of the system, $\vec{F} = \frac{d\vec{p}}{dt}$. Wait... isn't that assuming motion in space? Not necessarily. Some systems do have a meaningful definition of momentum that have nothing to do with moving in space. An electromagnetic example is inductance times current, $L\,i$, in a solenoid. And this looks a lot like $\vec{p} = m\,\vec{v}$, no? (except for the vector nature, or lack thereof)
So, if $\vec{F} = \frac{d\vec{p}}{dt}$ and $\vec{p}$ is proportional to $\vec{v}$ - let's say $\vec{p} = \alpha\,\vec{v}$ (I'm using $\alpha$ rather than $m$ because, again, I don't necessarily mean mass) - then
gain in KE = $\int_A^B \frac{d\vec{p}}{dt}\cdot\vec{v}\,dt = \int_A^B \alpha\,\frac{d\vec{v}}{dt}\cdot\vec{v}\,dt = \alpha\, \int_A^B \vec{v}\cdot d\vec{v} = \frac{1}{2}\,\alpha\,v^2\,\big|_{v_A}^{v_B} = \frac{1}{2}\,\alpha\,v_B^2 - \frac{1}{2}\,\alpha\,v_A^2$.
And there is your quadratic term, $KE = \frac{1}{2}\,\alpha\,v^2$.
From a very fundamental point of view one can see this with respect to Noether's theorem. Every symmetry is related to a conservation law, e.g. the symmetry of space with respect to rotational symmetry (space does not change if we rotate our coordinate system) results in conservation of angular momentum. In this framework energy is related to time symmetry.
In this sense all examples you gave are related to energy change with respect to some time dependent process. Kinetic energy is related to speed, which is related to acceleration, and the same is true for angular movement. If you look how the energy of a capacitor is derived, you use actually charge, which is brought on the capacitor using a current. Again a time dependent process. If you reverse time and require that energy stays constant, energy cannot depend linearly on a process that is linear in time. You can have a constant term, that is why I said "energy change", and the next term is supposed to be quadratic in time.
Note, you might have a fourth order term, though. Moreover, the argumentation is a little bit dangerous, as it somewhat uses the property it wants to explain as argument. If we reverse this, that would mean: A physical property depending on the sign of time, or not fulfilling time symmetry would not be called energy.
Also note, that all this goes out the window in general relativity, as time is connected to space and there is no pure time symmetry any more (and no conservation of energy either).
Is there any fundamental reasons seemingly many physical phenomenon has the energy in that form?
Yes. Think about a 10kg cannonball in space. If it’s travelling at 10m/s relative to you, you say it's “got” kinetic energy KE=½mv², and you say it's "got" momentum p=mv. Now brace yourself, then apply some constant braking force with your spacesuit jets, and catch that cannonball. Ooof, and you move backwards as you slow it down. Kinetic energy is looking at this in terms of stopping distance, whilst momentum is looking at it in terms of stopping time. That's it, that's all it is. The faster that cannonball is going, the more you move while you slow it down.
As an aside, note that you divide by c, which is distance over time, to go from energy to momentum. And that the measure we call momentum is conserved in the collision because your midriff and the cannonball shared a mutual force for the same period of time. The measure we call kinetic energy isn’t conserved, because some of that force was redirected into your deformation and heat and bruises. But nevertheless they are two measures of the same underlying thing. You cannot reduce the momentum of that cannonball without reducing the kinetic energy too. That's why we talk of energy-momentum.
