So just any kind of chicken with beer then?

Apparently in Korean saying "eat" instead of drink is very common, I think I've also come across "eat your medicine"

You can in Korea.

A lot of Korean words do this to shorten compound words.

If you wanted to spell everything out, sure - but 치맥 is actually a legit compound word for the Korean word "chicken" and "beer". For more information, I found the Wikipedia article enlightening: https://en.m.wikipedia.org/wiki/Chimaek

치맥 is a combination of 치킨하고 맥주

치맥 is a contraction of 치킨하고 맥주

치맥 is the contraction for 치킨 + 맥주. As the two are so commonly had together and sooo many diners cater to just that combo, its become its own word. Like 소맥!

this absolutely exists in korea