"Thee" isn't the informal form of "thou", it's the oblique form, ie "the bell tolls for thee", you'd never say "the bell tolls for thou", no matter how formal or informal.
In early modern English, thou and thee were both the informal 2pp, with thou being the subject pronoun and thee being the object pronoun. Thou/you is similar to tu/vous. Eventually, thou became archaic, and we think of archaic words as being more formal.
(Aside: In Middle English, ye was strictly plural, but ye became acceptable as formal singular as well, again paralleling vous. And as an additional aside, in the "ye olde English" phrase, the ye is actually þe, where þ is thorn, which wasn't available when the printing press came to England.)
