Univalence as a Principle of Logic First of all, thank you for replying to such a daunting topic. I think if more philosophers stuck their nose into mathematics, both fields would benefit.
Identification is a kind of pairing of two things. We identify one with the other. Isomorphisms describe how we go between the two things. Consider a sheet of paper. We could curl one side around to its opposite edge and 'identify' the two edges. Now we've got a cylinder. Another identification of the same two edges would be to twist the paper as we curl it and, voila,
Mobius strip.
Some theories have a bit of trouble (formally, at least) with identification because they consider two things to be equal just when their extent is equal. Two sets for example are equal iff they contain the same elements. The trouble comes in because, although two sets may be isomorphic (we have some idea of how to move between them), set theory can't identify the two because they contain different elements. Most mathematicians tend to ignore this I believe but HoTT and, similar theories, provide a way to formally identify isomorphic structures. It may seem like a technicality but getting this sorted out has advantages in making proofs and mathematics in general more computable.
This isn't a complete answer but I hope it clarifies more than it confuses.