There are hypothesis that dark matter particles are their own anti particle, which would mean that they would annihilate when they come in contact with each other.

The people looking for them are examining high energy bursts around supermassive black holes as potentially coming from that. No idea how they could ever approach finding any evidence for that. But that's their problem.

Dark matter interacts via gravity but not electromagnetism (including light). So its particles would have no electric charge, and thus no distinct antiparticles.

I think I remember hearing that it would react with dark matter the same way.

Like matter, it doesn't interact with dark matter to any detectable degree.

I say "doesn't" on purpose, because we have small containers of the stuff tucked away in labs, and it's (momentarily) created as an inevitable byproduct of particle accelerators and certain radioactive decays.

So my question is, does anti-dark matter exist, and what are it’s properties?

Nobody knows what dark matter is beyond it being everywhere and heavy, so double don't-know. There's no particular reason to think it would follow a similar pattern, though.

The problem with dark matter is that we don't know what it is. If it's a particle, then presumably there probably would be an antiparticle for it. However, we don't actually know if it's a particle or not.

There's some hypotheses that say that dark matter doesn't physically exist, and that our theories of gravity are just incomplete. There's some other hypotheses that say that dark matter might be really small black holes. And then there's some hypotheses that say that dark matter is a particle. Until we find out more about what it is, we simply can't say for sure what sorts of properties it has