*I am only trying to portrait the answer through mathematics
There exists a law used widely in electrical engineering, as I'm sure you have heard of it, the Ohm's Law, which states at a given R and I, the voltage could be calculated as V=IR. you need to understand that this law is not a linear equation, it's rather an instantons fact that stands, meaning at every differential time step, the product of resistance and current would give you the Voltage.
Now, the definition of power in physics is the rate at which work is done. in mathematical terms, that would be:
$$d(work)/dt = Power$$
So, lets now look into the primal defnition of voltage,work required per unit of electric charge, so it is essentially work/charge. If you look at this in differential terms, then it would become $$V = d(work)/d(charge)$$
thus
$$d(work) = V*d(charge)$$
so if you differentiate both sides with respect to time:
$$d(work)/dt = V*d(charge)/dt$$
(btw mathematicians would go berserk if they see you move differential segments around so don't do it around them!). So now we have the 2 tems we know in this equation:
$$Power = d(work)/dt$$
$$Current = I = d(charge)/dt$$
giving us the final format:
$$P=V*I$$
thus proving it represents something that's instantaneous. Now, think of voltage as pressure applied across a tube, and current representing how much water goes through, as you can see one is dependent on the other, so you cannot isolate one variable to say that the multiplication of both would rely on (meaning that I*V represents two variables not a constant and a variable). Now imagine these 2 equations representing the relationship between voltage and current:
$$V=0.5*I---------------------------(1)$$
$$V=0.75*I^2-------------------------(2)$$
so for (1), if you calculate power, it would indeed be proportional to I^2, however for (2), it will have a different relationship. Also look at Ohm's law like this that:
$$R(1) = 0.5$$
$$R(2) = 0.75*I$$
so for (1) R is constant and for (2), R actually depends on an external factor (the current itself in this case). This is a pure mathematical example and ofc does not hold in an experimental setup (as there are different factors influencing how the relationship would change)
So as a final line of answer, If the relationship between V and I is linear, then yes power is proportional to V^2, but this is not a generalized rule