Why opamps honor fourier series but not other series, for exmaple, power series?
Short answer: The complex exponential functions are eigenfunctions of the derivative and integral operators.

Some more explanation: The sine and cosine functions are somewhat more cleanly expressed as complex exponentials; that is, functions of the form
$$f(x)=e^{i\omega t}$$

using the identities
$$\sin \omega t = \frac{e^{i\omega t}-e^{-i\omega t}}{2i}$$
$$\cos \omega t = \frac{e^{i\omega t}+e^{-i\omega t}}{2}$$

And, you're probably aware, when you take the derivative or integral of an exponential function, you get another exponential out, with the same terms in the exponent. (This is what it means to be an eigenfunction of those operators).

So, if you design a circuit that performs linear operations (some combination of scalar multiplication, derivative, and integral, or a sum of several such operations) on an input signal, and you put a sine or cosine signal in, you will get a combination of sines and cosines out, at the same frequency.


  Is it possible to have circuits like opamps that respond to polynomial functions?


I don't know if it's impossible, but it would at least be very difficult, for at least a couple of reasons:


A polynomial function with a finite number of terms will eventually trend toward infinity beyond some positive and negative limits in time. But no real op-amp can produce a non-bounded output voltage.
The response would have to be non-time-invariant. Meaning the circuit would have to keep track of the time and respond differently at different moments in time. It would also have to do this for times before \$t=0\$, meaning it would have to predict the future to know when some event is going to happen that will designate the reference instant of time for the system.