Since family names are always unique shouldn't we be able to have any character name? It's so annoying having a name you really want but that along with all other spellings of it are taken :/

Yeah, this seems like a really unnecessary limitation. Because your family name is unique, the combination of your character name and family name is always unique. I guess that it could cause some issues in places where your character name gets shown without your family name, but IMHO that's kind of a weird feature anyway. I would have expected that all that would really matter would be your family name and that maybe in some places your character name would be displayed alongside it, but in general, the character name mostly just serves to help you organize your characters. Your family name is basically your username, and it's what people are going to use to identify you.

Trying using /invite $charactername command, you'll quickly realize why character names need to be unique.

And why is that even a thing? We already have family names to uniquely identify people. It really makes no sense to be referring to anyone by their character names IMHO.

In some cultures, it's overly formal if not plain rude to call people by their family name instead of their first name, maybe Pearl Abyss belongs to such a culture.

LOL. Your "family name" is your username. It's what identifies your account. The character names are just for distinguishing each of your characters. Getting annoyed by being referred to you by your "family name" in the game would be like being annoyed at being referred to by your username on any website and would just be silly.

Regardless, odds are that the fact that character names are unique is just some quirk of the implementation due to a decision they made early on, and they've just never bothered to fix it.