Thee and thou were once forms of address which were non-formal and intimate, hence the usage in the Bible and in wedding vows, so they would not originally have been used to denote respect to, say, strangers. However, they came to be used in literary works to indicate that the speaker is ill-educated and doesn't know when to use the correct form for the social situation; they are "overly-familiar", often due to being good-natured. This latter is probably the intent for at least some of the uses in Uncle Tom's Cabin.
The words can also be used in an insulting way, to suggest that the object is not worth a formal address.
As the words fell out of usage they became associated with old-fashioned speech and, as often happens, old-fashioned speech was itself associated with stiffness and formality. So a modern native English speaker will often have exactly the opposite understanding to the correct one - that "thee" and "thou" are especially formal instead of being friendly or relaxed pronouns for use in private situations with friends, children, and lovers.
A final complication is that regional (that is, English spoken in different parts of the UK) English contains these words even today but in my experience they are mostly used in the belittling/insulting manner but with an undertone of jokiness or teasing.