This question is probably trivial, for that I apologize.
Let's take a vector space $V$ equipped with a scalar product and a vector subspace $S\subset V$.
How many different projectors onto S, orthogonal with respect to the scalar product, can be constructed? And what if $V$ is, more generally, a Hilbert space and $S$ is a finite-dimensional subspace?
Intuition tells me that there should be only one, but I am not sure that is correct.
Start by ignoring the scalar (or inner) product, and fix a subspace $S\subset V$.
All the projectors onto $S\,$ (which means that the image equals $S$) are in 1-to-1 correspondence with subspaces $K\subset V$ such that that $S\oplus K=V$. The mutual relationship between projector and complementing subspace $K$ is determined by $K=\ker(\text{ projector })$. This holds true in every vector space $V$.
In the lucky case where a scalar product is available for $V$, a distinguished choice for the complementing subspace can be made, namely the orthogonal complement $S^\perp$. It is uniquely determined by $S$.
It follows that there is exactly one orthogonal projector onto $S$, thus confirming your intuition.