There are 3 playable characters currently in the LEGO Mario theme (Mario, Luigi, and Peach).
The app makes a two player mode available. Is it possible to play with all three characters in the app?
I ran into this issue recently. Unfortunately, this is not currently supported in the LEGO Mario app. Any two characters can be used together (even matching characters), but you can't connect more than two characters simultaneously.
This was clear in the app update that added support for Peach (emphasis added):
LEGO Peach comes with a bunch of new reactions and can be connected to either LEGO® Mario™ or LEGO® Luigi™ (sets are sold separately) to win bonus coins for teamwork.
Subsequent updates have not upgraded the multiplayer mode to support additional players.