If I walked up to, say, a Corinthian or an Athenian during the time of the poleis and said, "Hi, I'm (name omitted), who are you?" what would he tell me?
Yes. Or at least, why wouldn't you exchange names? If you look at instances of meeting in the Greek plays and epics, there is an exchange of names among unknown persons.
I'm not sure what else you are imagining?