I am a Cubs/Bulls/Bears fan, but there are a few other teams I root for, and players that I like. I think following players around to different teams is a more modern issue, since they are more likely to follow bigger contracts to new teams in the modern era (for better or worse). I'm a big Jimmy Butler fan, and I always root for him to do well, but I didn't necessarily root for the Timberwolves when he was there. I don't know how many Patrick Ewing fans became Raptors fans when he went there. I am/was a Jordan fanatic, and watched and enjoyed the Wizards when he was there, but I never really rooted for them.
Being exclusively a fan of one team can take away from the experience at times as well. For instance, for me, I'm not a huge football fan but root for the Bears. They haven't exactly had great playoff success in the last 4 decades, so I think that has, in some ways, contributed to me caring less about football in general. I might watch a good chunk of half the Cubs games through the season, which leaves little time to pay attention to other teams, and especially to any American League play (especially before the more balanced schedule started), so this past World Series matchup was between two teams I cared little about and new even less about, so I was really just enjoying it for the love of the game and the players I knew from their previous teams.
Different strokes though.