Name for equivocal similes such as found in hip-hop lyrics?

I'm fairly certain that's just called a pun*. Because it's particularly strained pun, you might want to call it catachresis. I don't think there's a specific term for it just because it's used in a simile. If there were, we might as well have a specific term for puns that are used in metaphors as well, and one for puns used alliteratively, and one for puns about chickens. With a limited vocabulary, words can only be so specific.

The defining feature of the category of puns you mention isn't that the pun is in a simile, but that that simile is a part of an allusion, generally with "attitude" and sarcasm associated with it. If you want a phrase that's been used before, you could try "biverbal allusion," which occurs in an 1833 essay by Charles Lamb.

*(Or double entendre, quip, witticism, calembour, paranomasia, antanaclasis...they're all basically the same; call it what you will.)

This sounds more like a syllepsis or a zeugma to me, wherein a word gets roped into doing double-duty. See this question for some possible differences between those, plus this surprisingly detailed Wikipedia article. The section that discusses when a "distributed term changes meaning" is particularly relevant to your State Farm example.