I've come across this sent: "He`s got short dark hair, glasses, a beard and A moustache".
Why there's "a" before "moustache"?

I thought it means both halves of them

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Nobody ever wears half a moustache :)
We say:
He has black hair
But:
He has a beard
He has a moustache

For some strange reason we treat 'hair' as an uncountable noun, like 'water' or 'mashed potato'. So we say 'some hair', 'much hair' and we don't use 'a'.
'He's going bald. He doesn't have much hair left'
'Some of your hair is going grey'

By contrast, if you want to talk about one or two individual hairs, you can use 'a', 'the' 'many'
'There is a hair in my soup'
'You have some grey hairs'
'Some of the hairs in your beard are brown'