Firstly, "thou"/"thee" is not modern English outside of dialectal usages (which I believe is its context in Uncle Tom's Cabin).
Secondly, you're right about the parallels to the two pronouns in Spanish, but "you" is the polite, formal option. Actually, Spanish is a bit more complicated, because it has (I understand) familiar and formal versions for both singular and plural. English is more similar to French; in both languages, the singular ("thou", "tu") is also the familiar, and you would use the plural ("you", "vous") to an individual if you wanted to be formal.
(Related reading: "T-V distinction".)
Thirdly, having said "no", let me now change it to "no, except when it is". Thanks to "thou" falling out of usage, most people are only familiar with it in old works, especially the King James Bible and Shakespeare's plays (both Early Modern English, from around 1600). The former, in particular, has caused "thou"/"thee" to be associated with formal situations, because God is addressed as "thee" (because it's singular, not because it's familiar).
However, that only applies to people affecting old-fashioned usage. They are quite likely to use "thou"/"thee" as a formal pronoun. People who use them because it's part of their dialect will, I'm sure, be using them with their original (singular and familiar) meaning.