VERY IMPORTANT! PLEASE READ:This is not a guide, but an important note to anybody trying for this achievement
On both the achievements app and the milestones page of Battletoads arcade, you will see this achievement and it should have a percentage (%) under it. Everytime you complete a playthrough with a different character, the percentage should go up 33% or 34% in this order:
0% (no playthroughs completed yet), 33% (1st), 66% (2nd), 100% (3rd and final)
However, many of us have completed the game multiple times but our % are stuck behind.
For example, I completed the game 3 times with all three toads but the percentage says 33% (only one playthrough completed), on both the milestones page and the achievements app.Proposed Solution
Whenever you've completed a single playthrough, hold
to load up the pause menu, and select Load>New Game, and start your new playthrough on a fresh save file. I can confirm this worked for me. Should work the same for you!