That’s a weird argument. So if the University of al-Qarawiyyin stops operating, then some other place becomes the first university?

No, just saying that considering the development of institutions of higher learning, there's a good chance that the first university and the oldest continuously operating one are one and the same. Not that it necessarily implies that they are one and the same.

That’s more of a modern understanding of what a university is like (and I doubt it is an accurate description of the mosque that Fatima al-Fihri made).

My point here is simply that Plato's Academy never developed into an institution like that. I'm sure that if it did, none of us would begrudge Plato the honor of being the founder of the world's oldest university.

A ‘community of masters and scholars’ one might say, or in latin ‘universitas magistrorum et scholarium’ which is the origin of the word ‘university’. The original point of Universities was the same as that of the Academy. The teaching of students is just a side effect of achieving that goal since teaching is a great way to learn.

See, here I have to disagree. The point of universities was to provide a community of masters and scholars, but explicitly for the access of the less-educated in order to perpetuate the knowledge possessed to the next generation, in the same spirit as guilds held masters and journeymen alike. All of the major institutions recognized as universities that I'm aware of (admittedly only a few, it's not my main field of interest) starting out with this more 'school-like' structure, rather than the structure of a round-table of academics. I know several developed out of more traditional church schools.