It is also called Paronomasia. It´s a play on words that produces a humorous effect by using a word that suggests two or more meanings or by using similar sounding words with different meanings. The humorous effects depend upon the ambiguities the words involved.
According to the Cambridge Dictionary, a pun is defined as "a humorous use of word or phrase that has several meanings or that sounds like another word." In statement, we can see that the speaker mentions that they laughed "at the double meanings" supporting the idea in the definition.
