What is the name of our Solar System?
what is our Solar System called rather than "the Solar System"?
I've found "Sol System" and "Monmatia", too, but is there more to it?
It's just called "the Solar System". (Plenty of places and objects have names like that; it's no different from "the Arctic" or "the Moon" or "the Sun".)
("Sol system" is an invention of science fiction writers; it has no general use outside some science fiction contexts. Anything else is going to be something similar, or general crackpottery of one kind or another.)
It should probably be noted that 'Solar' is a adjective form of Sol, just as Polar is to Pole. And that Sol is just the latin name for the Sun.
The solar system was named long before we knew that other solar systems existed. Just like the Sun is not the only Sun in the universe but we still call it the Sun.
It's interesting, because there is actually a lot of disagreement about the general use of the word "Sun". In Astronomy, usually Sun means only *our* star. Others claim that Sun can, and should, be used to reference any star nearest the planet, citing historical use in fiction. A lot of people also use 'Sun' interchangeably with 'star'. The actual meaning becomes somewhat ambiguous due to this.
To answer your question succinctly, the Solar System also goes by the names: The Copernican System, The Heliocentric System, and The Planetary System, in addition to the ones you have mentioned. There aren't too many other names, actually, so just stick to Solar System since it's the most widely accepted.