Is it possible to recognize when an endomorphism of a finite dimensional vector space is unitary for some choice of inner product?

Sure. An operator $T$ is unitary iff there is an orthonormal basis with respect to which $T$ is diagonal with eigenvalues of absolute value $1$. So, $T\in GL(V)$ is unitary for some inner product iff it is diagonalizable with eigenvalues of absolute value $1$ (just pick an inner product which makes a basis of eigenvectors orthonormal).