I am a bit confused about the definition of fine moduli spaces. As far as I understand, the difference between fine moduli space and coarse moduli space is the existence of universal family. For fine moduli space, suppose that $U\to M$ is the universal family, then given any family $X\to B$, there exists a unique morphism $B\to M$ such that we have a $B$-isomorphism $\alpha: X\to B\times_M U$. Is $\alpha$ require to be unique as well?
On one hand, I feel that the requirement of $\alpha$ to be unique is kind of unnatural. When we consider $M$ as a representable functor from the category of schemes to that of sets, it is natural to send a base $B$ to the isomorphic class of families. On the other hand, here is a proof from nlab that shows the $j$-line is not a fine moduli space. I don't see where the contradictions arises if $\alpha$ is not required to be unique.
In general, the modular curve $X_0(N)$ is not a fine moduli space for any $N$. I wonder if the fact that $-1$ is an automorphism of any family has anything to do with this result.