Baton Passes happen after triggering an enemy Shadow's weakness. The default console button is RT. After selecting a party member (or your character) the achievement will unlock.
You first gain this ability after entering the Castle with your core crew (Ryuji, Ann, and Morgana) for the first time after you get the timer to change Kamoshida's heart. To be more clear, not the first time on your first day, not the second time when Ryuji unlocks his, not the third when Ann tags along and unlocks hers, the fourth time when you are freely able to enter and explore the Castle freely.
The Tutorial Baton Pass does not count, you have to do this in a battle you initiate.