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Ex-tattooist here. Whether you asked to see ID or not (in case you didn't, make sure you always do going ahead. I used to photocopy it and attach it to the part of the form I kept for my reference, had to be passport or driving licence, no bus cards, student uni cards etc, only legal gov photo ID) the burden of proof may be on her to prove you didn't otherwise it is your word against hers on that.

But, the whole "I know a friend" is always a red flag. Oh don't they always just know a person? 😂 It reads very scammy particularly if you did not receive payment in the first place and she already had other tattoos that you have no idea if they were done by a different artist or a friend. Our piercer used to do aesthetics too, botox and filler. One time, some travellers came in, one wanted botox then later claimed she got a massive infection. Piercer was obviously mortified and asked her to come in to see her because "weirdly" they refused to seek any direct medical help and just insisted a full refund. They came in, big oversized sunglasses, when piercer asked to see the damage so she could advise if it was just inflamed or to tell her to see a doctor, she very quickly dropped the glasses then raised them again and there was nothing there. No inflammation, no infection. Turned out this pair were doing the rounds, getting free cosmetic work done then demanding a refund and additional compensation so as to not "take it further". Some people are chancers and sounds like you just met some of them. Might even be that the mother in this situation is just the lass you tattooed.

Ask them to come to the shop to talk it over. Tell them you are securing legal advice and may need to get the police involved. Gauge the reaction. As a mother though, if my son was tattooed underaged, I would go direct to the shop and speak to the artist face to face. There's something about having that anonymity of hiding behind a faceless text that sort of flags this as even more scammy.