There are polynomials at two layers.
This layer describes the elements of the non-prime field:
For example, for the prime $p=5$ and $n=3$, the elements of the field of
of order $p^3$ are given by polynomials of degree upto 2 ( 1 less than 3)
with coefficients in the prime field of order 5.
There is way of adding them and multiplying them remaining within the set, and satisfying all conditions that define a field. The elements are not number and not exactly polynomials either (rather their equivalence classes). As elements can only be 'polynomials' of degree a most 2, multiplication might yield higher degree polynomial outside the set: then we use a 'mod' arithmetic, that is polynomial division with remainder, yielding a polynomial of degree less than 3 as the 'product'. Let us agree to denote elements of these field by polynomials $h(t),k(t)$ etc.
In this layer there is a specific polynomial (for each elliptic curve) of degree
3, call it $f(X)=X^3+aX+b$. The elliptic curve (excluding the point at infinity) consists of pairs of elements from the field, $(h(t), k(t))$ such that
$k(t)^2 = h(t)^3 + a h(t) + b$. The point is simply denoted $(h(t), k(t))$.
As familiarity breeds contempt soon one starts writing $x, y$ for elements
of the non-prime field, not bothering to give a special notation for them as polynomials.