My comparative religions teacher says that Muslims believe that Jesus was the only man since Adam to be born free of sin. I thought that there was no doctrine of original sin in Islam. How can every man be born with sin, then? And was Muhammad? And is he (Jesus) the only man since Adam to be born free of sin?
You are correct in what you thought, in Islam there is no such thing as original sin. Every human being are born on pure human nature without sin, it is only when a human being comes of age that he/she become accountable for their actions.
From what you mention in your question, your teacher is mistaken. It is possible that he confused some concepts, for instance the creation of Isa/Jesus (peace be upon him) without a father is likened to how it is no more greater than the creation of Adam without a mother nor father.
the basic concept of sin in islam is that every prophet is a normal human being just like other human beings so they have abilty to sin. But Allah protects them from sinning. they are not just born sin free. Prophet Muhammad , Jesus, adam, all previous prophets were human beings like us, but the only difference between us and them was that they were protected by Allah from sinning.
And Allah knows the best