And why did Dumbledore say, at the end of the book, in his office ...'it means you're still a man.' Was there a possibility that he'd be anything else?

My guess is that there is an unspoken "as opposed to a monster, like Voldemort" in there. Harry is still a thinking, feeling, caring individual, and thus-- a man. I think Dumbledore fears that Harry will somehow head down the same path as Voldemort, so this is significant for that reason as well.