Yes, but a "fan club" especially one run by the team is a little different from a booster club.
The Florida Marlins & the Miami Dolphins have a "fan club" in which you pay your money to the team and get a cap and various other perks for your membership dollars. Probably the best one is the Dolphins who offer priority seating at practice, plus special receptions, autograph opportunities and the like. The Marlins one is kind of weak, compared to that.
A booster club is not actually related to the team and is a separate entity, although they should work in conjuction with the team.
The main purpose of a fan club is to give the fans something extra to keep them happy.
The main purpose of the booster club is to the the players the extra help they need when they are with the team.
I am reminded, BTW, of the Everblades Kids Club of the second season, which never took off.