The coach's fight with Leukemia has definitely been a unifying point, but this team was the team that Peyton Manning played for with all those other guys. Heck, I think more people paid attention to Peyton on the sidelines than the rest of the team on the field last year. Even though Luck is a big name in his own right, this team is a much more of a team concept and though you can't blame Manning for that, can you blame the other players for being distracted by the constant worry about his health?
You must be thinking of Reggie Wayne, because not many other players from the Manning years are left. The ENTIRE offensive line is brand new (including TEs), The ENTIRE defensive line is brand new, only a handful of linebackers remain (Angerer, Freeny, Mathis) and a handful of DBs (Bethea, Powers). Donald Brown is still around, but I've never liked him and he hasn't been much of a replacement for Addai.
This team has been completely overhauled.
There was plenty of recognition to go around when Manning was still there... there were a lot of Pro Bowlers on those teams, and they had name recognition. The stadium wasn't completely filled with Manning jerseys on game day, everyone had their favorite player. Maybe the reason you think of the Colts now as more of a "team" is because you'd be hard pressed to name any of the players on it other than Luck right now.