It is easy to see that an orthogonal projection on a separable Hilbert space is positive.
A completely positive map $T$ is one for which every tensor product
$$T\otimes \operatorname{id}_{\mathbb C^{n \times n}}$$ is still positive.
Thus I was wondering whether this is true for an orthogonal projection as well?
It looks to me that you are confusing two things. A projection $p$ on $H$ is positive as an element of $B(H)$, since $p=p^*p$. It is positive in the sense that it is selfadjoint and its spectrum is nonnegative.
But when you discuss complete positivity of a map $T$, you need $T$ to positive as a map (i.e., in the sense that it maps positive elements to positive elements). So the domain of $T$ is a C$^*$-algebra, or more generally an operator system. Such notion makes no sense for an orthogonal projection, since you have no positivity in the Hilbert space.