I've read that $Rel$ (the category of sets and relations) is a 2-category by considering 2-morphisms to be inclusion of relations. Is $Set$ also a 2-category by considering 2-morphisms to be inclusion of functions (as relations)? I can't find the reason why this is not true.
Give two sets $A,B$, and $f,g:A\to B$, if $f\subseteq g$ as relations, then $f=g$.
So this is a trivial 2-category - the only 2-maps are the identity maps - but it is a 2-category.
So it is a $2$-category in exactly the same way that any category can be made a $2$-category.