I have zero interest in this game. The more that comes out about it, the less interested I am. I was also really turned off by the PC Gamer preview. I typically do not play MP either. The only time I do is when a game is made with MP in mind (L4D, Battlefield, things like that) and even then, I mostly devote time to my SP game collection.
That said, this game could have some cool MP. I wouldn't mind a scavenger hunt mode where people are racing each other to find treasure. That could be cool. They could inhibit each other with traps or special abilities that are character specific, or have abilities that give them an edge in some situation. Perhaps Lara can run a little faster while someone else can climb faster or jump a bit further. You could use this same idea and have animal hunts. The match creator (because we all know it will be P2P because that works awesome all the time, right) can define a certain weapon to be used and a specific animal. You can even have some of the people play as other predatory animals in the competition. You then have the prey spawn in a random area that is specifically hard to get to so the characters have to use some of those aforementioned special movement abilities to get to the prey faster (hopefully) than others will.
All of this is, in my mind, very specific MP that would make sense in the context of the game. It's a Tomb Raider game so scavenger hunts and animal hunting makes a lot of sense. All of that said, I'm sure it will be some tacked on contrived MP where people are killing each other. I mean, I never played the Uncharted MP, but I recall reading it was just like Deathmatch bullshit.