You can't know that there's only one named sol by the locals.

One day we'll be sued by aliens for trademark infringement.

If that’s where we’re going I’ll try to pedant-raise you.

Assuming the cosmological principle is true and the universe is infinite, wouldn’t we be guaranteed an infinite number of Sols? ;)

Do people actually call the Sun "Sol"? I thought that was more of a video game thing.

The Romance languages use that name. English uses "Sun", but just as it borrows a ton of stuff from Latin/French/etc., it also borrows "Sol" for its word "solar".

Also, Captain Archer in Enterprise used the name Sol when making contact with aliens.

And in Germanic mythology the personification of the sun is Sól.

In PIE it's sunnōn. In some languages that evolved to some variation of sun or son, in some to became sol (notably Latin). And many use both variations in some capacity

Latins and medieval scientists did. In Italian we call it "Sole".

If you speak Portuguese or Spanish, yes.

I can't think of any English-speaking places that do... but you see it used in "solar", for example.

It's Солнце (Solntse) in Russian.

We live in what's known as a planetary system. Star systems involve only stars.