It's the job of foreign media organisations to manage a university's PR/marketing?

No but if they promote some but not others, then they're implicitly taking up that job.

Is it reasonable for an university to become familiar for the general public across the world?

Yes, absolutely. It's how we cross distances and learn more about each other. It's how we become more than the sum of our parts.

There are many thousands of universities. They can't all be familiar. Those that most consistently produce quality research (or have the best PR) will be well known. Getting there? That's a bit of a chicken or the egg problem: Media outlets aren't going to want to bog down a headline with unrecognizable names-- fewer clicks. Arguably worse for the research too since. When it is noticed, it will be linked better to the University, but it won't be noticed as much because of a title that isn't SEO friendly or doesn't grab attention as well.

There are thousands of sodas, and you know what Coca-Cola is.

And I'm familiar with many universities in my own country as well as a handful of the most prominent ones abroad, so..?

I think we’re in agreement, haha.

I don't think that contradicts what I wrote-- it's an example of it.