Azalea1012:
IF Tae was the one Shinwoo liked in MS, maybe he didn't directly give the bracelet to Tae? Like maybe he left it on his desk or in his locker with a note but didn't reveal who he was? Maybe the note said something sweet or inspiring so Tae kept the bracelet even though he didn't know who it was from? Shinwoo may have told his friends he bought a bracelet and was going to/did give it to a boy but they didn't know who. They started bullying Shinwoo so he never revealed who he was to Tae or who the boy was to the friends?
I do think it's odd that after bumping into Shinwoo he immediately wants to make friends. Maybe on some level he did remember Shinwoo (subconsciously) so after bumping into him that day it made Tae have that desire.
This is impossible, why? Because Tae and Shinwoo met in highschool, not middle school. Someone explained it in details in the comment section which makes more sense to me than Tae finding a bracelet someone left with something sweet written on it and wearing said bracelet everyday without bothering to find out who it was from.
Also, you make it seem as if Tae only wanted to make friends after meeting bumping into Shinwoo. That's unlikely because they've bumped into each other many times and Tae didn't even remember or notice Shinwoo. Tae had also been thinking of changing way before he met Shinwoo and official noticed him and his first thought was that he wouldn't someone like him for a friend.
Shinwoo is already winning, why is it so important that we create stories for him that connects him to Tae.