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Yes, to "give a shit" or a "damn" means to care marginally more than not caring at all.

He's using it in an ironic[1] sense to mean that they really do care. This is similar to BBEdit's tagline[2], "it doesn't suck," which, read literally, doesn't mean it's good.

[1]: contrasting with expectations and norms

[2]: http://www.barebones.com/products/bbedit/index.html

You’re 100% correct in your interpretation of the strict grammatical meaning.

I think maybe ‘not giving a shit’ existed as a turn of phrase first, and ‘giving a shit’ is sometimes used (incorrectly in the grammatical sense, but ok in the colloquial sense) as the complete negation/opposite.

Thankfully, Ismo explains it all: https://www.youtube.com/watch?v=igh9iO5BxBo

Thank you for this moment of humour. It was much needed.

You're not wrong at all