What makes a cooperative game require multiple players?
The biggest thing I can think of is limited knowledge. Hanabi is a cooperative game where each player has access to different information (no one can see the cards in his or her own hand, but can see everyone else's cards).

As soon as different players have access to different information, playing by yourself becomes impossible. The other examples I can think of are partnership card games: Bridge, Pinochle, Canasta, Tichu, etc. They all have strict rules against communication between partners, so you cannot let your partner know what cards are in your hand. Due to this rule, a person could not end up playing both sides of the partnership by himself.

This type of thing could easily be implemented in non-card games as well, as long as there is some information that some players have but other players don't.

The other way that one could make a co-op game actually require multiple people is to have rules against table talk. Even if all knowledge is open, you can make it illegal to discuss moves. For example, you could play a Pandemic variant where any discussion between players about what should be done is illegal. You are still allowed to ask what cards other players have (or just play open-handed; same thing really), but no other discussion allowed. Thus, each player needs to decide what he wants to do on his turn all by himself, and the challenge comes on picking up the slack where other players are lacking.

Damage Report took another approach. All players play simultaneously and asynchronously (not everyone is on the same turn), and there are enforced time limits to turns. This prevents the game from being played solo (though a dominant player can still influence others)

"Hidden roles" type games can also work this way. Battlestar Galactica, for example. You could consider this a co-op game, because most of the players are all working together. Though instead of fighting against the game itself, as with Pandemic, you are fighting against the 1 or 2 other players who are not on your team. But due to the fact that roles are hidden, each player only knows his own role. This makes it impossible to play by yourself. Of course most probably wouldn't consider this a co-op game, but the basic principle is still there.. most of the people playing the game are playing co-op with each other.