I don't think it should be obvious. If you could measure spectra, that would tell you that starlight and sunlight are the same, but you could still think they were very tiny suns that were near. You would need to measure parallax to know they were far away. Neither of these are possible without precision technology, though you could probably argue that it could have been done in ancient times with enough effort.

Maybe if the solar system had more than one star, or there were other stars very close, people would have caught on a lot quicker.