When did astronomers realise that the stars were similar to the Sun? I'm not asking for when this was established, but when also the hypothesis was first proposed.
As found here, Anaxagoras suggested around 450 BC that the stars are far away Suns. This is the first known person to suggest this.
As is fairy common in the sciences, the idea was reinvented numerous times, but it didn't really catch on until the mid-1800s when spectral analyses on the Sun and stars carried out by Fraunhofer and Secchi (see this page) started showing remarkable similarities between Solar and stellar spectra.
Aristarchos of Samos was the first to talk of a heliocentric system. He certainly put the sun at the center, though he puts stars on a sphere.
You (King Gelon) are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.
His writings have been lost due to the great cleansing of pagan documents by christianity.
You have a good question. The answer is - we dont know. We know for sure it happened long time ago, probably at the beginning of science as we know it. You should know that there is no MATERIAL EVIDENCE about any ancient scientist. All text which are presented in posts above are just "official" hypothesis, in reality it could have been very different. Look here http://en.wikipedia.org/wiki/Antikythera_mechanism I would guess a guy who did that for sure had some clue about what stars are.