In a lecture, our professor defined an allowable surface patch for a surface $S \subset \Bbb R^3$ to be a diffeomorphic surface patch of $S$.
But is is possible to have a diffeomorphism between an open subset of $\Bbb R^2$ and an open subset of $\Bbb R^3$?
If $f: U \subset \Bbb R^2 \to V \subset \Bbb R^3$ is a diffeomorphism, meaning that $f$ is a differentiable invertible map with a differentiable inverse, then the differential of $f$ at some point $x_0 \in U$ is a linear map $Df(x_0): \Bbb R^2 \to \Bbb R^3$, and it is invertible as its inverse is given by the differential of $f^{-1}$ at $f(x_0)$, hence there would be an isomorphism between $\Bbb R^2$ and $\Bbb R^3$, which is impossible. Isn't this right?
Indeed, an open subset of $\mathbb{R}^2$ cannot be diffeomorphic to an open subset of $\mathbb{R}^3$, for the reason you outlined. In fact there cannot even be a homeomorphism, by invariance of domain.
But a surface is not an open subset of $\mathbb{R}^3$. Consider the standard sphere $$S^2 = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1 \}.$$ Hopefully you agree that it is a surface, and indeed $S^2$ is locally diffeomorphic to an open subset of $\mathbb{R}^2$. But $S^2 \subset \mathbb{R}^3$ is not open.