A popular assumption about black holes is that their gravity grows beyond any limit so it beats all repulsive forces and the matter collapses into a singularity. [...] Is there any evidence for this assumption?
It's not an assumption, it's a calculation plus a theorem, the Penrose singularity theorem.
The calculation is the Tolman-Oppenheimer-Volkoff limit on the mass of a neutron star, which is about 1.5 to 3 solar masses. There is quite a big range of uncertainty because of uncertainties about the nuclear physics involved under these extreme conditions, but it's not really in doubt that there is such a limit and that it's in this neighborhood. It's conceivable that there are stable objects that are more compact than a neutron star but are not black holes. There are various speculative ideas -- black stars,
gravastars, quark stars, boson stars, Q-balls, and electroweak stars. However, all of these forms of matter would also have some limiting mass before they would collapse as well, and observational evidence is that stars with masses of about 3-20 solar masses really do collapse to the point where they can't be any stable form of matter.
The Penrose singularity theorem says that once an object collapses past a certain point, a singularity has to form. Technically, it says that if you have something called a trapped lightlike surface, there has to be a singularity somewhere in the spacetime. This theorem is important because mass limits like the Tolman-Oppenheimer-Volkoff limit assume static equilibrium. In a dynamical system like a globular cluster, the generic situation in Newtonian gravity is that things don't collapse in the center. They tend to swing past, the same way a comet swings past the sun, and in fact there is an angular momentum barrier that makes collapse to a point impossible. The Penrose singularity theorem tells us that general relativity behaves qualitatively differently from Newtonian gravity for strong gravitational fields, and collapse to a singularity is in some sense a generic outcome. The singularity theorem also tells us that we can't just keep on discovering more and more dense forms of stable matter; beyond a certain density, a trapped lightlike surface forms, and then it's guaranteed to form a singularity.
Why can't some black holes be just bigger neutron stars with bigger gravity with no substantial difference except for preventing light to escape?
This question amounts to asking why we can't have a black-hole event horizon without a singularity. This is ruled out by the black hole no-hair theorems, assuming that the resulting system settles down at some point (technically the assumption is that the spacetime is stationary). Basically, the no-hair theorems say that if an object has a certain type of event horizon, and if it's settled down, it has to be a black hole, and can differ from other black holes in only three ways: its mass, angular momentum, and electric charge. These well-classified types all have singularities.
Of course these theorems are proved within general relativity. In a theory of quantum gravity, probably something else happens when the collapse reaches the Planck scale.
Observationally, we see objects such as Sagittarius A* that don't emit their own light, have big masses, and are far too compact to be any stable form of matter with that mass. This strongly supports the validity of the above calculations and theorems. Even stronger support will come if we can directly image Sagittarius A* with enough magnification to resolve its event horizon. This may happen within 10 years or so.