Do we call other stars "suns," or is that just the name we gave to our star in our solar system?
The terms sun and star are interchangeable.
We tend to use the word sun for our star and the word star for others. Though it is also common to refer to a planet's sun for exoplanets.
It is quite complicated as most stars have several names and our Sun doesn't have an official name. Some people give the word sun a capital letter to make it a proper noun when referring to our Sun.
Stars can have actual names and another name indicating where it lies in a constellation. For example the star Rigel in the constellation Orion is also known as Beta Orionis.
It is further complicated by the fact that there are several star catalogues and a star can have a different name in each.