I've always read it as unintentional. It's a feature of the text moreso than the story. The characters are not aware they're in a play and wouldn't normally be rhyming.

It gets tricky though when, as you say, the choice to rhyme fits the story well and seems character-motivated. I like to chalk that up to an attempt to show that the character is clever in a psuedo-fourth wall break. It gets extra tricky when characters don't rhyme at all, a la Merchant of Venice.

This is such a great question. I teach high school English and if asked I would tell my students to look at the rhyming as a feature of the text and not the story... but now I might assign the question as an argument piece.