How and when did humans discover that our sun is a star
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It depends on what you mean by “discover” and “star”.
The ancient Greeks figured out by the 3rd century BCE that the sun was a gigantic source of heat and light which was located very far away from Earth. They also worked out that moonlight was reflected light from the sun, which explained moon phases and eclipses. Eratosthenes used trigonometry to calculate the approximate circumference of the Earth and extrapolated that to approximate the distance to the sun. Around the same time, Aristarchus of Samos proposed that the Earth and other planets orbited around the sun as a solar system.
Although the 3rd century BCE philosophers Democritus and Epicurus proposed that stars were similar to our own sun, these were purely philosophical ideas without scientific support. The concept that the sun and stars were comparable objects did not reach mainstream scientific consensus until the 17th century, when improvements to our understanding of gravity allowed astronomers to better understand the scales of celestial motion. However, even at this time it was still accepted that the sun was the center of the universe.
The heliocentric model was not completely rejected until the early 20th century, when observations by Harlow Shapley and Edwin Hubble proved that there were innumerable distant galaxies outside of our own Milky Way. It was briefly believed that the Milky Way formed the center of the universe, and that all other galaxies orbited around our home galaxy, but Hubble quickly disproved this model as well by measuring the red-shift of light from distant objects to chart the expansion of the universe. However, I would consider the rejection of heliocentrism to be the final straw when humans scientifically established the Sun was a star just like any other, and not a uniquely important object in the cosmic scheme of things.
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