I'm going to be vague on the timing because it depends on what you mean by "determine". Which in turn depends on what is meant by "know" in science. There is a very good model for stars which predicts that balls of plasma that are prevented from collapse by fusion in their cores will look like the sun up close and will look like stars from a distance. So people believe that both the sun and the stars have the same nature. However, while there is direct evidence of nuclear fusion in the sun (neutrinos) there isn't the same evidence of fusion in stars. So it can be argued that this hasn't been "determined", and given the nature of scientific discovery, might never be.
It also depends on what you mean by "the same nature". What if we suppose that two things have the same nature, but are completely wrong about what the nature of the object actually is. We didn't have any real understanding of the core of stars until the 1930s, and we are still filling in the details.
The early claims that the sun is a star were based on general evidence and some incorrect, or at least odd beliefs. Later discoveries showed how starlike the sun actually is but there was no Eureka momement; there is no single event that you can point to in which someone discovered this.
Anaxagoras in the 400s BC suggested this or something like this. He suggested that the sun was a ball of molten metal, and the stars were fiery stones, so something of "the same nature" He proposed that the sun and the stars were torn from the Earth, and then ignited. He probably supposed that meteors were "falling stars" and might have linked meteorites to meteors. He almost certainly believed that the sun was bigger than the stars. So "of the same nature" but the sun was the greatest of the stars. He is significant because he is probably the first to give a physical theory for the nature of an astronomical body. He made the first steps to bringing the study of the heavens within natural philosophy, or "physics".
Anaxagoras was the first to hypothesise that the Sun and the stars might have the same nature.
The notion that the sun is not special is suggested by Copernicus, and there were mutterings about it in the period after Copernicus' publication; it is implicit in some of Galileo's work. However, it was Giordano Bruno who gave full weight to this idea. His evidence seems merely to have been that, if the sun was very distant, it would appear as a star. The observation was naked eye viewing of the sky. He does not seem to have estimated how distant the stars would be, as he had no way of measuring the actual brightness of stars of the sun, but he most clearly declared that:
“The composition of our own star and world is the same as that of as many other stars and worlds as we can see.”
(And by "our own star" he means "the Sun")
It wasn't until the middle of the 18th century that any actual evidence for stars being of the same nature as the sun was found. Angelo Secchi used spectroscopy to analyse starlight, and found that it was very similar to sunlight. By this time though Secchi was confirming the orthodox position rather than proposing a new discovery.
With the measurement of stellar parallax, another piece of the puzzle settled into place, now that the distance to stars could be measured, the were found to comparable to the sun in brightness as well as in spectrum. And developments in nuclear physics in the '30s-'50s showed how stars and the sun could be powered by similar nuclear engines in their cores.
So this question doesn't afford a simple single answer. This wasn't a sudden paradigm shift. Rather it was a gradual process. However, if you need just one name to hang on this idea then Bruno would seem to be the first to clearly assert that stars have exactly the same nature as the sun.
(Much taken from http://www.astronomytrek.com/who-discovered-the-sun-is-a-star/)