Not the phrase "Time flies like an arrow" on its own, no. (A time fly isn't a thing that exists, I'm pretty sure.) But paired with the "Fruit flies like a banana", and knowing my father's sense of humor, yes, that is the joke.

The meaning of the words change based on how you read them.

Paraphrased from my textbook:

Time (noun) flies (verb) like an arrow (adv. clause) = Time goes fast

Time (verb) flies (obj.) like an arrow (adv. clause) = Track the time of flies the way you would track the time of an arrow.

Time flies (noun) like (verb) an arrow (obj.) = Insects known as time flies are fond of an arrow.