How did the term "Mistress" take on two rather different connotations?

One meaning of the word is "female master." The Latin equivalent would be Domina.

Another connotation is "lover." Not quite what one associates with "Domina."

Or was there a connection between the two made when men took "dominant" women as lovers?

Could one possible instance of this phenomenon be the medieval "courtly love?"

https://history.stackexchange.com/questions/1125/what-was-courtly-love-amour-courtois

The Oxford English Dictionary’s earliest citations for mistress are from the fourteenth century when it meant, very broadly, ‘a woman having control or authority’. It had the sense of ‘a woman loved and courted by a man; a female sweetheart’, with no indication of impropriety, very early, perhaps a mere 100 years after its first recorded use. Almost from the start it seems to have had multiple meanings. That’s hardly surprising given how common a word it must have been. The dominatrix sense, however, doesn’t appear until 1921.

Check out the online etymology.

This term comes from French, as many English terms do, and in French, words vary in gender depending on whether they apply to a man or a women.

A man in charge of a number of domestic staff, workers or pupils would be called "maître" and a woman "maîtresse" (same word but applying to a member of the fair sex).

A woman came to be called "maîtresse" as a romantic metaphor for the influence she held on a man's heart (at that time, this term did not imply -or reject- illicit relationships or even a carnal relationship), just that the woman was loved. The term then came to designate a fiancee and in the 17c a women having sexual relations with a man outside of marriage.

In short, no, the term Mistress did not originally imply a dominant lover and still does not by default, unless the context suggests it, of course.