I think there should be 2 separate "shelves" on the team overview page. One would contain all the major awards the TEAM has won (league titles, cup titles, division titles and, why not, maybe also PL titles for supporters). The other one would contain all the INDIVIDUAL awards that any players have won while playing for the club (MVP awards, ROY awards, top scorer, top rebounder, etc.)
I'm pretty sure some teams would soon have so many trophies to show that the overview page would take probably too long to load. However, I think there's an easy solution: just show the, let's say, 10 most important awards per category, and then have a link to another page which would only show these trophies. All of them :).
And, of course, the player's history page should show all the individual awards the player has ever got. That is an addition I want to see implemented asap, to be honest.