Seriously...
I got LA Noir last night and the game is awesome but for some reason I can't figure out how both my brother and I can play the game without mixing up our games. I can't find any save slots or player profiles, it's like the game doesn't want more than one person to enjoy it, all there is is a new game or continue, it's ridiculous, and it doesn't work with steam family sharing either.
If anyone knows a way for both of us to play the game without buying 2 copies please let me know, I'm guessing I'm not the first one this has happened to.

You can't. I had this problem when I was playing the game with my girlfriend.

PC games today are designed to be a one person, one licence thing. Most games on GOG are an exception to this rule, and Steam is taking some steps to alleviate this problem somewhat with Family Sharing, but most games with UPlay, Origin or other proprietary account-based DRM systems are a one-person affair. GFWL is another exception as it does allow secondary offline accounts.

The best way to be able to share a copy of LA Noire is to buy the PS3 or Xbox 360 version.

Edit: What you could try is creating a new Windows account for your brother so that his save files are added there instead of yours. This might not always work as the saves may be bound to an account, but it's worth a shot. I don't recall this working for LA Noire - I simply got prompted to enter the serial again and was told under no uncertain terms that it was already registered (i.e. the game's DRM activation was bound to my own Windows account)

I barely remember LA Noire, but I think there's profile option somewhere in the main menu. I'm sort of remembering (or misremembering) the Social Club entry in the main menu.

I don't really know as I never tried it. I know that when my brother played the game using my account, he used the CASES option to start any case he wanted. I wasn't in the middle of any case, so nothing got overwritten.

Eh, games like that I generally hunt down where the save data is being stored. Snag the relevant save file/s, put it in a new folder called "Game Name - Saves" or "Save Backups for Game" or whatever (stored wherever you like), and start a new game; since new game generates new save data. Then when I want to switch saves, I just swap the files. Ta da, multi-user slots. Sometimes the saves are in Documents, sometimes in Library/Application Support, sometimes in the .app itself (note: I'm on a Mac, ymmv). It's also helpful for games which start struggling if you have too many saves. For instance Neverwinter Nights really slows down if you have 25+ saves, so throwing all the ones you aren't using right away in a folder outside of the app helps a ton (same thing with Civ V and Crusader Kings II, etc).

You can try to run installed game via Sandboxie. You create one sandbox for your you, one for the other person and save files you creare should not interfere.

I don't think so, but I'll check to be sure later.

Check this out.