I'm not familiar with this variant, but it is still NP-complete for any fixed $p$.
Given a graph $G$ and an integer $c$, connect to each vertex $v$ a clique $C_v$ on $(p+1)c-1$ vertices.
If the original graph $G$ has a valid coloring $\chi$, then we can color the clique $C_v$ as follows: the color $\chi(v)$ appears $p$ times, and all other colors appear $p+1$ times. You can check that every vertex has exactly $p$ neighbors of the same color.
Conversely, suppose that the new graph has a coloring $\chi$ in which each vertex has at most $p$ neighbors of the same color. This is only possible if each color in $C_v$ appears at most $p+1$ times, and so some color $\chi'(v)$ appears $p$ times, and the rest appear $p+1$ times. This implies that $\chi'(v) = \chi(v)$ (since otherwise $v$ would have $p+1$ neighbors with the same colors), and furthermore that $\chi$ restricted to the original vertices is a valid coloring (in the usual sense).