The sun is considered a star because it has all the characteristics of one. In fact, the sun is unremarkable. There are countless stars with the same attributes as the sun throughout the universe.
Early philosophers considered the sun to be special. Its power was obvious: The sun illuminated the streets and caused temperatures to rise. They often considered the sun to be like the moon in many ways. The movement of the stars was baffling, and the planets caused even more questions. The special nature of the sun seemed evident to philosophers and astronomers, who believed that it rotated around the Earth. When later models showed that the sun was the center of the solar system, the sun seemed a bit less remarkable and special.
During the Renaissance period, astronomers began proposing that the stars were actually distant bodies like the sun. Because the sun is so bright and powerful, they reasoned, it must be visible over an incredible distance. Because of this, they realized that the sun would look like a star from an appropriate distance. Over time, this view became dominant, and the work of William Herschel helped solidify the sun's position in the Milky Way galaxy as one star among many.