Yes, two ways. One, look at them under a microscope and try to identify at least family/genus based on morphology. Two, sequence the DNA.
That's not very helpful for most people, though. So beyond that, it takes some training to be able to identify major families. It's rather similar to mushroom identification.
The good news is that the vast majority of mold out there is not poisonous. Many will make the cheese taste bad, though. I've never gotten sick from eating rinds, and I've eaten some crazy cheese experiments over the years.
