The guys with suits are indeed coaches.
All those guys in the polos are managers. I know a couple on Michigan. Believe it or not those guys on the bench don't even make up all the managers. There's also one guy recording the game, plus about 4-6 that don't even get to sit with the team due to bench space.
They probably don't need so many during the game, but they do give a lot of specialized tasks. Two years ago my friend's job was to write the play name for the players during the timeout. At CMU they have a kid whose job is just to track the type of fouls each ref calls. Another simply tracks whether he used the correct mechanics.
I know both my friends were up until 4am in Brooklyn a couple weeks ago after the Oregon win putting together scouting film and a report that would be ready for the coaches in the morning. They also use a lot of guys for rebounding during pregame, to help during drills, and basically make everything "appear" during practice and games so the coaches can focus solely on coaching.
Honestly, I don't know why they have so many compared to other sports, but I would guess part of it is due to attrition. I know quite a few kids quit MSU/UM each year just because it's such a huge time commitment. Hockey and football are generally weekend only, with two or three distinct start times, so there's a routine that you can get into. Basketball, on the other hand, has games any day of the week during the non-conference season, with about 10-12 possible start times. Even when conference season hits the only regularity you have is that you know you won't be playing Monday or Friday, but all others are fair game. That's only Big Ten too, most other conferences still play seven days a week.