And the French term for it is idiotisme (not sure how the English version was derived), which carries with it the idea that it is something not normal (or perhaps not "logical"), but which of course a native speaker understands.

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

The relationship between "idiot" and "idiom" goes back to Ancient Greek, according to etymonline.com (idiom, idiot), and it's the same "idio" as "idiosynchratic", meaning particular to the individual. (Just as well I checked, actually -- for years people told me that "idiom" was from the Ancient Greek for "language", and I nearly repeated that here.)

The French "idiotisme" is probably pretty recent (I don't know of a French equivalent to etymonline), as part of France's demonisation of common speech and regional variations.

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A language belongs to its native speakers, and when you speak it, you are a guest in their homes.
If you are not a good guest, you have no right to complain about receiving poor hospitality.

It's a term I didn't know but I've looked it up (in French) - it means an idiom or turn of phrase that is specific to a language or dialect and that can only be translated by an equivalent, so it can't be transposed directly word for word from one language to another. Béarlachas or un anglicisme as it's known here, would be the opposite of that.
I'm sure there are many expressions in Irish that get lost in translation.
(I looked it up at the Larousse.fr online dictionary, but if you want etymologie there's the CNTRL data base, sorry if it's OT)

Anyhow, the boundaries of the term "idiom" vary depending on your definition.

Most people think of "an idiom" or "idioms" in terms of clichés, proverbs, particular phrases etc. However, I've seen "idiom" used as a collective/uncountable noun to describe any rule which doesn't follow directly from the grammar.

Under this definition. "tá sí agam" vs "I have it" is a difference in idiom, and similarly "I am 111"(be) vs "j'ai 111 ans"(have) is a difference in idiom. (NB: not "an idiom" here, just "idiom")

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A language belongs to its native speakers, and when you speak it, you are a guest in their homes.
If you are not a good guest, you have no right to complain about receiving poor hospitality.