> So how do we know if something is a singularity?

We do not. It's a strong prediction of general relativity because inside a black hole, all possible trajectories lead towards the center (or a ring for rotating black holes): Trying to stay at a finite distance is as impossible as trying to stay in Sunday.

> If I’m hearing right, yes some neutron stars could gain enough mass to become black holes

Yes, because they collapse above a certain mass. They are not neutron stars any more if that happens.