Very strange grammar question about club/fanclubs

could a grammar expert please help me?

One is as good as the other, not that I claim expertise.

A fan club (usually two words) is a group of people who admire something or someone. It sounds like the club being proposed (The we miss ... club) is just a club or group whose common feature is missing someone. It is therefore not a fan club. Maybe you should find some online friends who are not stupid dickheads? Are you aged about 12?

W: This will sound very very weird for a grammar question I know...
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First, W, it isn't a question of grammar. It's merely word choice.

W: I am in this groupchat on Whatsapp and a friend of ours is in the hospital and a friend of mine changed the name to: We miss .... Fanclub. And then I told him fanclub didn't make any sense. I told them the correct term was The we miss ... club. Because a Fanclub is for people like Michael Jackson or something.
So I told them if we change the name to The we miss ... club then it would be correct but because we're participating in an activity (in this case missing ....) it should be club because a club is a group where people participate in a activity together. And now they're all making fun of me because they all say I'm incorrect. So which one is correct?

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Both are correct because they are examples of you people using language to say something you want to say. Even if fan club is normally reserved for a celeb doesn't mean it always has to be.

Using fan club could illustrate a greater sense of caring.

My cat is my biggest fan.

(He ignores me, but not as much as he ignores everyone else)

Well, if they admire the ill person, it can be a fan club.

Personally, I'd call it a fan club, injecting a bit of light heartedness into it.