Paintball guns do have triggers and look similar to pistols or other firearms, though they are very different in terms of how they function. A paintball gun is definitely not a firearm as it is fired by compressed air (making it an airgun) and not with combustion (though the short-lived Tippmann C3 does qualify as a firearm). That means that firearms laws and ordinances don't apply to paintball guns, but, since they are air-powered, airgun laws and ordinances do apply.
The problem with the term "gun" is that it typically refers to something lethal, which paintball guns most definitely are not. They look like guns and operate with a trigger, but so do glue-guns.
To avoid any confusion, many people refer to paintball guns as paintball markers. Since the term "marker" is less known than "gun", I refer to them as paintball guns on this site.