You are right, it's conflating "lion" and "lying" as they sound similar. Certain accents in the UK will also pronounce "lying" closer to '"lion", kind of like "lyan", missing out the "ng" sound altogether.

There isn't anything more to it. It is very difficult to explain why "giraffe" is a good choice here; humor hard to analyze or explain. "Giraffe" is funny because 1) it sort of goes with "lion" because they are both African animals of the plains; but 2) it is so bizarre and different from a lion that it is funny to imagine anybody confusing them. The joke would be less funny if the bartender said "It's not a lion, it's a tiger." The pun is the same but the incongruity isn't as great. I think there is also something about the sounds of the two words "lion" and "giraffe" that work well, but that is like trying to analyze poetry.

It isn't a very good joke, tbh. I think it being a giraffe helps a bit, as the mental image of a fallen giraffe is more vivid and 'fallen' than that of a fallen octopus or hyena, but I still don't think it would make anybody actually laugh.

It does not have to be a giraffe. It is just a play on words because just as April said, lion and lying sound similar.

You can use other animals, like rhinoceros, and it's still funny. It's probably best if you use a wild animal from Africa. It makes the joke work well.