Back in the day, you would say the Morning Sun or the Evening Sun depending on which of the two daily editions you were talking about. I think when it became a single edition paper, folks started calling it the Sun paper.
The Sunday Sun as well. Over the 186 years that one variation or another of these miserable publications have been in print, it seems likely that "The Sun Paper" became accepted locally as mention of a singular version of the several different and published Sun versions.