Is it always possible to factorize $(a+b)^p - a^p - b^p$ this way?

It is true in general. These are often called Cauchy polynomials — see, for example, Fermat's Last Theorem for Amateurs, Chapter VII. The power of $a^2+ab+b^2$ is either $1$ or $2$ as $p \equiv -1$ or $+1$ modulo $6$, respectively. There are many proofs in the literature (Cayley, Glaisher, etc.).

The polynomials $P_n(x)=P_n(x,1)$ are called the Cauchy-Mirimanoff polynomials.

They are defined for all $n$ natural $n\geq 2$ by: $$ (X^n+1)-X^n-1 = X(X+1)^E(X^2+X+1)^eP_n(X) $$ Where $E=0$ if n is even, and 1 otherwise, and, as @KierenMacMillan notice, $e= 0,1,2 $ according to $n= 0,2,1\mod 3$.

We know that $P_{2p}(x)$ and $P_{3p}(x)$ are irreducibles when $p$ is prime. It's still unknown whether all Cauchy-Mirimanoff polynomials are irreducibles...

For example, the proof that $P_{3p}(x)$ is irr, use the fact that those polynomials are self-reciprocals $(a_0=a_n, a_1=a_{n-1}, a_2=a_{n-2}, ...)$, and the degree of $P_n$ is even; this ensures the existence of a unique polynomial $P_n^*$ called the "reciprocal transform" of $P_n$, concretely, if $deg(P_n)=2d$: $$P_n(x) = x^{d} P_n^*\Big(x+\frac{1}{x}\Big)$$ The irreducibility of $P^*$ and $P$ are directly linked through Dikson's Theorem, which states (in the case of the Cauchy-Mirimanoff polynomials, with even degree) that $P$ is irreducible $\Leftrightarrow$ $P^*$ is irreducible. Finally, Netwon polygons are used to proof that $P^*$ is irreducible.

The details can be found here: https://trace.tennessee.edu/utk_graddiss/707/