I'll call a family of polynomials "binomial theorem-respecting" if they satisfy:
$$ f_m(x+k) = \sum_{i=0}^m {m \choose i} k^{m-i} f_i(x) $$
For example, the family of polynomials $f_m(x) = x^m$ satisfies the above. Slightly more generally, the family of polynomials $f_m(x, c) = (x+c)^m$ also satisfies the above.
I've found a more complicated family that looks like:
$$ \begin{align} f_0(x, c) &= 1\\ f_1(x, c) &= x\\ f_2(x, c) &= x^{2} - c\\ f_3(x, c) &= x^{3} - 3 x c\\ f_4(x, c) &= x^{4} - 6 x^{2} c + 3 c^{2} + 2 c\\ f_5(x, c) &= x^{5} - 10 x^{3} c + 15 x c^{2} + 10 x c\\ f_6(x, c) &= x^{6} - 15 x^{4} c + 45 x^{2} c^{2} + 30 x^{2} c - 15 c^{3} - 30 c^{2} - 16 c\\ ... \end{align} $$
For $c=0$, this is the original family $f_m(x) = x^m$.
Are there more families? Is there a general construction of them?
Hint: Polynomials $f_m$ which fulfil \begin{align*} f_m(x+y)=\sum_{k=0}^m\binom{m}{k}f_k(x)f_{m-k}(y) \end{align*} are said to be of binomial type.
[2022-11-14]: Thanks to the comment of @Cam.Davidson.Pilon I'd also like to point to Appell sequences. These are polynomials of the form \begin{align*} f_m(x+y)=\sum_{k=0}^m\binom{m}{k}f_k(y)x^{m-k}\tag{1} \end{align*} This identity (1) is called Appell Identity. It is stated as Theorem 2.5.8 in Umbral Calculus by Steven Roman where this and related topics are thoroughly treated.