Puns also bear similarities with paraprosdokian, syllepsis and eggcorns.
From the en.wikipedia.org
There is a rhetorical figure found several times in the song called syllepsis, in which a word is linked to two or more other words in different senses.
From the orlandosentinel.com
A syllepsis is a particular kind of zeugma, and there is a clear distinction between the two in classical treatises written on the subject.
From the en.wikipedia.org
Alternatively, a syllepsis may contain a governing word or phrase that does not agree grammatically with one or more of its distributed terms.