Is ENet thread-safe?

ENet does not use any significant global variables, the vast majority of state is encapsulated in the ENetHost structure. As such, as long as the application guards access to this structure, then ENet should operate fine in a multi-threaded environment.

Isn't ENet just re-inventing TCP?! What's the point?

In a perfect world, that would be true. But as many have found, using TCP either in lieu of or in conjunction with UDP can lead to all kinds of nightmares. TCP is a good, solid protocol, however it simply isn't up to the task of real-time games. Too much of TCP's implementation dictates a policy that isn't practical for games. If you want to use TCP, then do so – this library is for people that either don't want to use TCP or have tried and ended up being discouraged with the performance.