To put it simply:
Saving in multiplayer modes is disabled, splitscreen being one of those things. The only exception to my knowledge is SRB2 Top-Down. However, choosing a Sonic & Tails file (or using a file with two characters) allows you to save your progress, at the cost of having one camera.
The Sonic & Tails option, contrary to what some may think, does support co-op using Player 2's controls. However, there are a few quirks with this option, being that you can take control of the bot using P2's controls, Player 2 will respawn (as a bot) near Player 1 if they go too far, and Player 2's controls are relative to Player 1's camera. This is mentioned directly in Sonic & Tails' character selection text.
The other option requires modifying the source code, as SRB2TD did. I'm not an expert in source code modification, however, so I don't think it would be an easy feat.