Yes, you can't simply point to a component and say "pole" or "zero" -- apologies to Polish readers, of course
-- it is a concept that is several levels of abstraction down.
Example:
Physical: you can point at a resistor (component).
Abstraction 1: a resistor has resistance (electrical characteristic).
Abstraction 2: resistances combine with inductances and capacitances to make complex impedances. Complex impedances vary as a function of frequency.
Abstraction 3: when a rational approximation (i.e., a ratio of polynomials) is used for this approximation, we can factorize the numerator and denominator. When this is done, the factors are called zeroes (numerator) and poles (denominator).
Abstraction 4: optionally, we might analyze how the poles and zeroes vary as a function of circuit parameters (e.g., component value, amplifier gain, etc.). In this case, we apply theorems like the Routh-Hurwitz stability criterion.
As long as you are capable of grasping abstractions -- this shouldn't be challenge for you. A journey, certainly; there is a lot to learn here!
In general, any analysis, done at this low level of abstraction, works in the mathematical domain of polynomial factorization. It's a notoriously difficult class of problems, so the mathematical tools are difficult to use, and the numerical results are typically unstable (i.e., small changes in inputs sometimes cause large changes in outputs). As engineers, that's okay for us; we're fine with "close enough", or "tweak until it's right".
In that case, the hardcore mathematics isn't needed, and an easier lesson helps inform us what adjustments we should be targetting, so that the design isn't solved in a single step (which might be possible, but quite difficult), but rather, evolved until a close enough, practical solution is had.
Tim