Are you 100% sure your ex didn't know and didn't encourage it? How much time is your DD spending with his new partner alone? Why is he not spending his time with her?
Firstly, men can be particularly in observant, and if the DD has long hair then it wouldn’t be that hard to hide them. After all, OP didn’t realise until the DD was sat on her lap. It wasn’t something she initially noticed either.
WRT why the DD spent time alone with the GF, why not? I’m sure the OP doesn’t spend 24/7 with her DD? This idea that time with the other parent is “contact time” and only he should be spending time with the child is ridiculous. Yes if he’s constantly palming off his DD on to other people then it’s an issue, a shopping trip with a GF who presumably lives there and is part of their lives shouldn’t be perceived as a problem or inappropriate. I’m referring to that in general obviously, without taking the piercing into account.
With regards to the GF having had her ears pierced, I would be fuming, but I would also not dismiss the idea that it was your DD’s wish and that she didn’t ask for it because she knew that you and her dad said no.
Before going in with all guns blazing I would get all the actual facts. I.e. did the girlfriend really suggest it or did she just agree to it? Because while both are inappropriate there is a difference.
Also, did the GF tell the DD to keep it a secret from her parents or did DD decide she could keep it a secret because she knew what her parents’ reaction would be. I personally suspect it was DD who decided to keep it secret because the GF, as a grown woman, would have known that she couldn’t keep it secret for ever, but DD is young enough to not have realised that.
But to suggest that an adult is encouraging a child to keep secrets from her other parent has potentially very sinister connotations, and if it’s not true then it puts a different spin on things.
You need to talk to your DD and be absolutely sure that she didn’t ask and the girlfriend just agreed.
The GF was 100% in the wrong,Even if she did only agree, as were the shop, although I’m not entirely sure how these shops enforce the “only a parent or guardian may sign,” law as it’s impossible to prove that you’re a child’s parent.
But I think you need to be led by what actually happened, rather than by your initial shocked reaction. IYSWIM.