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, it came from europe, and didn't often have the thorn character.
The thorn, when written in decorative script, looks a bunch like the letter y -- so a Y (often with a superscript e) is what they used instead in printing (also instead of separate t h, for space reasons), along with some abbreviations of other common words starting with th, like this and that.
So "ye" became an alternative printing for the word "the", understood as exactly the same thing. Put it on a sign, it's an abbreviation saving a little space.
It was also understood as distinct from ye, the plural pronoun[https://en.wikipedia.org/wiki/Ye_(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"