If I am not mistaken that is strictly the way baseball handles it (player deciding which team he represents).
Pro Football Hall of Fame they just wear a jacket for the ceremony. Then the Hall looks at the player's career (basically how many years with each team) and assigns that as the primary team.
Is a New Hall of Fame Member Enshrined as a Member of a Team?
Obviously, teams take great pride in the accomplishments of individuals who have been a part of their organization. Often individual teams and even the Hall of Fame will list enshrinees according to the team or teams on which they spent a significant period of time. An enshrinee, however, is not asked to “declare,” nor does the Hall of Fame “choose” a team under which a new member is enshrined. When elected to the Pro Football Hall of Fame, an individual is recognized for accomplishments as a player, coach, or contributor.
Hall of Famers who made the major part of their primary contribution for any one club are listed in bold. In cases where a player contributed about equally and/or in a major way to two or more clubs – he is listed in bold under both clubs (ie. Gary Zimmerman is listed with both the Vikings and the Broncos as his primary teams). Hall of Famers who spent only a minor portion of their career with any club are listed under that club in normal font.