When a free agent has multiple exact offers of any value at the ML level the tiebreakers are as follows: relevant IQs of the coaching staff and ballpark. If the players previous franchise is also bidding on him, then that franchise will receive a loyalty bonus.
When the coaching staff is used to break the tie position players will look at the hitting IQ of the hitting coach and the fielding IQ of the fielding coach. Pitchers will look at the pitching IQ of the pitching coach and the bullpen coach. Please note that an FA will sign with the team that has better coaches in hand, which may or may not be the team that ends up with the best coach and/or staff.
In cases where the offers are still tied and the coaching staff has been factored in ballpark is used to determine which franchise the player will sign with. Pitchers always want to pitch in a pitchers park, while position players will always want to play in a hitters park.
When a free agent has multiple exact offers of any value at the minor league level the tiebreaker is organizational depth at the player's primary position.