I think people already get it, but just in case... there are two names.
- login name: you type this when you log in (it is unique)
- display name: this appears alongside your posts (it might not be unique)
There is a long standing argument in user model circles on whether display names should be unique (where early members get nice names and later members end up being Greg435), or whether everyone should go by their chosen name (where people can have the same name with the associated confusion).
For better of for worse, we are doing the latter. But there are a couple of things that are exacerbating the confusion (one bug and one usability problem) that I will fix at some point.
As people have pointed out, the profile picture is the best way to have a unique signature that people recognize. And after all, a picture says a thousand words.