Why adjoint operator is not a generalization of transpose matrix?

Question

I know in finite Hilbert spaces, the adjoint operator is a sort of generalization of the transpose matrix, in the sense that if $M$ is symmetric, then $\langle Mx,y\rangle=\langle x,My\rangle$. So if $M$ is not symmetric, we define the adjoint $M^*$ as $\langle Mx,y\rangle=\langle x,M^*y\rangle$. So it's a sort of generalization of transpose matrix (it's the way we justify it in the lecture). But the assistant says that it's a good point of view in finite dimension, but it doesn't work in finite dimension (despite if its self adjoint), and I don't understand why. His explanation was to be careful with the domain, but I still don't understand.