It's a joke. It has an inside as much as any bottle does, but the only point of a Klein Bottle is the topological form so they're using topological language to talk about it.
I think maybe it's just semantics. I seem to be incorrectly assuming that the inside of the bottle is the space where the liquid goes. And the inside of the möbius strip would be the gap where you put your neck if you were to wear it as a necklace.
Would it be more correct to say that there is no inside surface? And it was just a pun that went over my head?
yes it does, you just pour in/out from the bottom.