Old English had the þ character (capital: Þ), used as a 'th' sound, from old Norse influence (where now only Icelandic still uses it).
When printing came about, the thorn was already in decline, so it didn't really make sense to add it as a distinct piece of metal type, it'd mainly make things more complex.
But also, space was at a premium, so making common words like 'the' shorter was still a decent idea.
And 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 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.
These days, thorn is long gone, so when you put "Ye olde shop" on a sign, people will easily pronounce the y as the vowel (yee).
And this wouldn't be an issue, except ye ('the') at the time was entirely from ye, the plural pronoun (similar to modern american y'all).
So when you pronounce that shop's name with 'yee', it's either mispronouncing the thorn, or gramatically incorrect use of the plural pronoun ("Y'all shop"), meaning you've sort of invented a third thing we pretend is old, but was never really part of the language.