After much hesitation I came to the conclusion that I have to accept the answer by Peter LeFanu Lumsdaine. Still, I decided to add something that helps me more to understand it.
It seems that the key is in the very notion of monomorphism. "Without homotopies" it is just that $m:A\to B$ is a monomorphism if $mx=my$ implies $x=y$ for any $x,y:X\to A$. Whereas "with homotopies" it has to be refined to this: any proof of the fact that $mx$ and $my$ are equal must be liftable to a proof that $x$ is equal to $y$. So we get something like $m:A\to m(A)$ being also onto to some extent. That is, some things in $m(A)$ must be liftable to certain things in $A$. This at least shows that one needs some more restrictive conditions on $m$ than just being a monomorphism.