The main reason you can drink a drink but not eat an eat boils down to grammar and meaning.
The word 'drink' can be both verb and noun. A verb is a word which describes action. A noun describes a person, place, or thing. A drink can be named Coca Cola, Sprite, espresso, Root Beer float, or any other beverage. Additionally, you and I can also drink any one of those drinks I have just named off.
As for 'eat,' we usually use this word as a verb. If we want to change this word into a noun, we can try saying something like 'Let's pick up some eats before we go to the show.' Here, 'eats' is the plural form of 'eat.' Depending on where you live (whether in America or another country), this can sound a little strange if you're not used to speaking this way. Most people would say something like, 'Let's pick up some food.'
So, to answer your question, a drink can be used as both noun and verb, but 'eat' is defined as a verb. To change the word 'eat' into a noun, we can pluralize it as 'eats,' but this is an informal or colloquial use of the word. Hope this helps!