It’s an old joke, and people are generally amused that they can’t think of a good answer. The guy in the comic has a good answer, though.
Of course English does have a couple synonyms for “synonym;” they just aren’t in common usage.
For example, you have “metonym,” a type of synonym where you refer to part of something to mean all of it. When you say “wheels” to refer to your car, that’s a metonym.
“Analog” can be used to mean “synonym. ” Not very often, though. I suggest you make it into a phrase, such as:
“Beater” is a noun analog for “car.”
Sorry for spoiling the joke.