English used to have the þ character (capital: Þ), used as a 'th' sound, and basically from old norse influence (where now only Icelandic still uses it)
When printing came about, the thorn was already in decline, mostly used for a few common words(verify). Printing also didn't have the thorn character and it made little sense to make it more complex.
Ans since the thorn character written in decorative script looks a bunch like the letter Y, Y with an (often superscript) e is what they used instead in printing (also instead of separate t h, for space reasons), also using this pseudo-thorn for the few abbreviations of other common words starting with th, like this and that(verify).
So ye became an alternative printing for the word "the", understood as exactly the same thing.
It was also understood as distinct from ye, the plural pronoun (similar to modern american y'all).
These days, thorn is long gone, so now when we see 'ye' in old print, people pronounce the y as the vowel, so 'yee'.
So enough uses of "Ye olde shop" has led to inventing a word, basically just through mispronounciation of an archaic thing from your own language.
It also doesn't make sense as the plural pronoun, because that'd be as gramatically incorrect as "Y'all shop"