No, there's only one solar system in the entire universe. There's countless star systems though, but only one of those stars is named Sol.
/pedant
/pedant
Assuming the cosmological principle is true and the universe is infinite, wouldn’t we be guaranteed an infinite number of Sols? ;)
Also, Captain Archer in Enterprise used the name Sol when making contact with aliens.
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