My partner came across this book from 1737 in her research, which mentions "innumerable orbs". This sparked my interest in whether there was a common belief in exoplanets in the 18th century, and when people even found out the sun is a star.
I've read around a bit and found that several people proposed this as early as 450 BCE. However, I'm less interested in wild guesses which happen to be right, and more in the evidence people used to nail this down as the commonly accepted view. It seems to be that hard evidence for this is difficult to get, as even the closest star is unbelievably far away.
This page gives an overview, but jumps really quickly from Newton to observations in the mid-19th century. I'm particularly interested in the period in between, given the source that sparked my question.
The idea of our Sun being a star is very old, dating back to the ancient Greeks several hundred years BC. Though that idea was displaced with Ptolomy's model throughout most of the middle ages. It was Bruno who revived/rediscovered the idea during the Renaissance, which Galileo also favored and by the time of Huygens and Newton in the 17th century most scientists believed that the Sun and the stars were the same type of objects.
Huygens went so far as to estimate the distance to the brightest star by comparing the brightness of a small portion of the Sun to that of Sirius. This is a remarkably clever experiment since areal brightness is unaffected by distance, and determining the size of the sliver of the Sun that is as bright as a distant star means that you can determine the apparent size of that star, and thus its distance. However, he did not know that Sirius is intrinsically much brighter than the Sun, which introduced a large error in his estimate.
Nevertheless, the general consensus in astronomy was that stars and the Sun were the same type of object from then onward. The absolute proof that this was so came slowly, and piecemeal. The first strong line of proof came in 1838 when Frederich Bessel measured the distance to a star via parallax. Knowing the distance and apparent brightnesses of stars one could determine their actual brightnesses, which proved to be similar to that of our Sun, showing that they were objects of tremendous luminous intensity. Additionally, several categories of variable stars were explained by invoking the idea of "starspots" from the 18th century onwards, which was a potentially tenuous but nevertheless real point of comparison between stars and our Sun.
The next line of evidence came in the 19th century as well, with the advent of astronomical spectroscopy. The spectra of stars and the Sun were measured for the first time, and discovered to be fundamentally similar. Later, during the early 20th century there was enough data to plot the temperature and luminosity of many stars, and the Hertzprung-Russel diagram was invented (around 1910). This showed that there were different populations of stars in different phases of their lives, and it also placed the Sun within that population, showing it to be fundamentally like many other stars in the galaxy.
Later, in the 1930s the source of the luminosity of stars was finally elucidated and discovered to be nuclear fusion. This was the last piece of the puzzle to cement our understanding of stars, at that point there was little room for doubt that the Sun was a star. Later advancements in measuring the age of the Earth and the Solar System via radiometric dating, resolving the disks of distant stars, discovery and measurement of solar neutrinos, and the advent of asteroseismology further coalesced the understanding of the Sun as a star.