You don't need to know meme grammar to learn English grammar
I think it doesn't make that much sense to evaluate this text for grammatical correctness. Or at least, it would not be worthwhile for an English learner to get confused about this: learning what's going on in this kind of meme is unlikely to help you with any other area of English. Memes don't ...
Context is important in English, and especially today, ideas and concepts are often conveyed very succinctly.
The artist in me cries out for design (Robert Frost)
This can be crudely summarised as
Me: Cries out for design
Therefore in a declarative sentence such as
Me: watches an entire season of a TV show in one night
The “me” above refers to ...