Perhaps the easiest counterexample comes from letting $X$ and $Y$ be countable infinite disjoint unions of points, and setting the vector bundles to be one-dimensional. Then you just do a dimension count.

A more sophisticated counterexample is the function $e^{xy}$ on $\mathbb{C}^2$. It is globally holomorphic, but it is not a finite sum of products of single-variable entire functions. You can check this by considering the restriction to $\mathbb{N} \times \mathbb{N}$, and writing the values as entries in a big matrix. The matrix is a union of nested van der Monde matrices of nonzero determinant, so it has infinite rank. Therefore, it can't be the sum of finitely many rank 1 matrices of the form $f_i^T g_i$

When one of the manifolds is compact, the isomorphism is a corollary of cohomology and base change. You can find a proof in pretty much any book on several complex variables. See, e.g., Grauert, Remmert *Coherent analytic sheaves*.