is there an example schemes such that underlying spaces are not homeomorphic but sheaves are isomorphic? Maybe if there exist, I want to see that example
Let $Y$ be the spectrum of $k$, $X=\mathbb P^1_k$ and $f: X\to Y$ be the canonical morphism. Then $f_*O_X$ is a sheaf supported in one point (that of $Y$), so it can be identified with $(f_*O_X)(Y)=\Gamma(X, O_X)=k$. Therefore $O_Y\to f_*O_X$ is an isomorphism, but $X$ is not isomorphic to $Y$.