Can you explain the relevance? If I put N items randomly into >> N buckets the chance of there being a second item in a particular bucket is small (as opposed to there merely being a bucket with two items, as in the birthday "paradox").
That doesn't apply here, since the birthday paradox is about the existence of a collision, not that any particular sequence collides.
Most people in the room will still have unique birthdays even if one pair share theirs.
I guess you get (infinite?) many results which all have the same hash and one (or more) of them will be shorter than the hash.