Does every solar system have its own sun?
Sun system is a star system. Sun was called Sol in Greece and Italy. So, Sun system is also called solar system.
In the recent millenniums (and also now), the Sun God is called Surya or Aditya in India, Apollo, Sol, Hyperion or Helios in Greece and Italy, Aton in Egypt, Tonatiu in Mexico, Ten Suns in China, Amaterasu in Japan, ..
Every star-system, including our solar system, has orbiters ( in different planes ) around the star,
The star is at the common focus of all the elliptic orbits,
Every solar system has at least one sun.
The terms sun and star are synonymous. We tend to use the term sun in the context of a planet. A planet has a sun which is itself a star.
A solar system is a star with at least one planet orbiting around it.
Many stars have one of more companion stars. Hence many solar systems will have two or more suns.