Why doesn't my controller respond when I start a PS1 game?
When you launch your PS1 game, press "SELECT + B". The "Quick Menu" will then open for you. Go to the options, then disable the 2 "Multitaps".
Once done, go back twice (so twice "A"), then select "Quit Retroarch".
Restart your game and everything should be in order.
Small precision: you can later reactivate the Multitap 2 to play with up to 4 players (for shock team races, for example).