In the UFC, Muay Thai seems to be much more common than kickboxing, karate, or tae kwon do. Is this actually true, or is it just my perception? What's the relative breakdown of the styles?
What are the reasons for this?
It's almost impossible to shoehorn fighters into individual styles these days since they're all training multiple martial arts. Rarely do we see the old school "karate guy" or "judo guy". Even those experts (Stephen Thompson, Dan Kelly) are also experts in several other martial arts. Tony Ferguson (UFC lightweight title contender) even says he trains every martial art he's ever heard of. To answer if this is true: yes, most modern MMA fighters train Muay Thai.
Now why do they all train it? Muay Thai uses more tools than the other striking arts. Elbows and knees are huge additions. The other striking arts generally stick to punches and kicks, whereas elbows and knees are an essential part of MMA. You can't throw most punches and kicks effectively in extremely close range, but that's where elbows and knees shine.
The biggest difference of all is THE CLINCH! Who's more comfortable in a clinch is often what decides fights. Even an elite wrestler is going to have an awful time taking someone down if he can't survive in the clinch. An elite kickboxer will get murdered if he can't hold his own when someone wraps up both hands around the back of his neck for a Thai clinch. Fighters with clinch control dictate where the fight takes place; the open floor, against the cage, going down to the mat, etc.
All that Muay Thai specific stuff being said, Muay Thai also excels at open floor striking. Good ol' "stand and bang" style fighting. Muay Thai has been adapted for MMA to aim for maximum damage strikes rather than more traditional point scored Muay Thai. This makes for some absolutely brutal leg kicks, and knees/elbows meant to finish fights.