I don't think people hate either, we just don't treat them as actually-true when using language. Nobody ordinarily means to include "skin" when they say "organ" (possibly doctors or academics in certain contexts, but it's not normal usage), and nobody means "Afroeurasia" when they say "island".

It's a kind of reductio based on what people will tell you when you ask them to define "island", to show that definitions they come up with tend to have outcomes they don't like—happily, we may just ignore all that and keep using the words in a useful and mutually-intelligible way, so that language remains tenable. In fact, the reductio isn't proving people wrong when they say "afroeurasia isn't an island". Nobody treats it that way, so it's not. If they did, we'd need a different word for "island".

Africa, Europe and Asia are continents, so can’t be Islands

The definition of continent is very fuzzy and mostly man-made. So all you're really saying is Afroeurasia isn't an island because we decided it isn't.

Yup. That's how language works: Words mean what they mean because we've decided that they do.