Yes, the progress is automatically saved and synced to the Steam Cloud individually - so when you next join a co-op game it will have stored which levels you have completed. This also means that your progress is shared across any computer you play Portal 2 on.
It's worth noting that progress is never saved during a test chamber though, only upon successful completion.
You will be able to pick which level to start at next time you start the game - if you go to the specific chapter you want to play and use the buttons until you've selected the level you want, then go into the assembly machine as usual.
Contrary to all the answers on here, it is in fact possible to reset your progress in co-op. Make sure you are the host (you invited your friend, rather than you were invited by your friend). Start a co-op game. Make sure you are in the hub when doing this.
Warning: This applies to both players. Make sure the other player is fine with you doing this.
Unfortunately, this doesn't work with ss_map. You have to be in a session with another player, and this will set both players' progress.
Enable the developer console in the options, then press ~ to open it. If you are having trouble, see this question. Then, enter the following command in the console:
mp_mark_all_maps_incomplete
You might see a weird message about DLC, but the command will have no other visible effect. Now, enter this command:
changelevel mp_coop_lobby_2
This will reload the lobby, and your progress will be reset at zero.
You can use this command to change your progress manually:
mp_mark_course_complete <course>
Whereas the number is the number of the course you want to unlock (have finished), minus 1. So to unlock course 1, use mp_mark_course_complete 0, to unlock 2, mp_mark_course_complete 1, and so on. This means that mp_mark_course_complete 0 unlocks course 1 fully, and will then open 2.
There is also mp_mark_all_maps_complete if you want to get Professor Portal.
Thanks to fail badp♦ for allowing me to waste his time.
Best Answer
I don't think you can do what you ask, but unless you want to keep saves to access extremely specific parts of levels, you really don't need any more saves. Once you've finished a level you can always go back to it afterwards and play it again. So if there are some puzzles you like, you can skip straight to them (or to the beginning of the chapter, anyway).
Addendum from comments: Remember, you can always quicksave (F5) and quickload (F9), which is much faster. But you always overwrite your previous quicksave, so be aware of that. However, due to the design of the game it's essentially impossible to lose any progress short of a complete wipe of your saves.