Before I hung up I decided to check and make sure that they were actually looking for me. I asked for the full name on the account and all they had was a first and last name. The collections agent then proceeded to tell me this person's social security number(!) and birthdate, neither of which matched my own. So, I was able to confirm that I was not that person and he said he'd forward the account to management with a note that they'd found the wrong person.
I called their office a little while later to let them know that he'd told me the social security number and birthdate of one of their customers. If I'd actually recorded any of that information it wouldn't be hard for me to impersonate this person especially since we share the same first and last names.
I need to remember to check my credit report sometime soon to make sure that this doesn't show up there.