What category of AI would a 2048 bot be considered?
I would initially say that you can classify 2048 as a MDP problem.

For this reason, I would consider looking at Reinforcement Learning (an area of Machine Learning). 

Another option, however, more specific would be to look at the performance of Monte Carlo Tree Search.

There is a great discussion in StackOverflow about 2048 different approaches have been tried:
https://stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048