What about Plato’s Academy? It was an institution of higher learning, and it certainly existed long before the one in the meme. I’m not saying that the Academy was the first one either, it wouldn’t surprise me if there were others before hand.

Certainly, if we want to get into the question of who exactly was first, we could probably come up with a few dozen names. That it is very early in the academic tradition is what's important to the meme and the message.

It seems to me like the claim “the oldest currently existing university” got converted to “the first university” and those are definitely not the same.

I would say "yes-and-no."

Whatever university has the strongest claim to be the oldest currently existing university probably also counts as the first university, because of the irregularity of the structure of learning institutions in the Classical and pre-Classical era (to the best of our knowledge). Plato's Academy, which you mentioned, was arguably more of an enduring round-table than an institution with students and teachers, and to our knowledge didn't have any sort of curriculum. Plato's Academy was a place where philosophers and important folk got together to mutually intellectually enrich each other, rather than to pass on specific knowledge or competencies.

Generally we consider a university more like a school - with set staff, set students, and a set course of learning; Plato's Academy had a head, I believe, but not regular teachers or students, and simply discussed matters in a learned way amongst themselves. If you went there ignorant, you would learn something, but attendance and learning, or attendance and teaching, were not synonymous; and you came and went according to when you were satisfied with what you were discovered, rather than there being courses or degrees that could be completed.

China might have a good claim on the oldest university, tbqh, but I couldn't trace the history there, personally. It's far from my fields of interest.