Your wife holds German Citizenship, and as such can apply for an ESTA. You’ve done this (well done for answering the questions truthfully) and it’s been approved (as it should be).
German law does not usually allow dual citizenship (except for EU/EEA nationals, and Iran is not in Europe).
I’d prepose that your wife gave up any claim for Iranian citizenship when 12 and took German citizenship.
Iranian law says once an Iranian, always an Iranian citizen.
Iranian Law does not supercede International law, including UDHR Article 15 – the right to change citizenship. At least from an US perspective, which is what counts in this context.
The ESTA is sufficient, as your wife is a German citizen.
This is tricky, as shown by the discussions in the comments.
The question is: which law do the US consider here. And I would say it is a safe bet to say that the “once Iranian always Iranian” law would not be seen as valid by the US IO, since there is no corresponding paperwork. As far as I can see from your description, your wife does not have a dual citizenship. This is the most important question here, since dual citizenship with Iran as one of the two requires a visa. Although I’m sure they’d need some sort of proof of citizenship.
It is another question whether the US consider being part of her mother’s passport as having been issued a travel document by Iran, but I’m sure they wouldn’t check this far back.
A last caveat: ESTA only means approved to travel to, not necessarily approved to enter the US. I’ve personally never had this issue, but my former citizenship is from an ex-Soviet country, so it’s probably not a hot issue right now when compared to Iran. One never knows, so be prepared, just in case.
An ESTA is part of the VWP (Visa Waiver Program). With a valid ESTA a visa is not required for visits that comply with VWP requirements, as visiting family does.
whether she still needs a tourist visa because she was born in Iran and Iranian law says once an Iranian, always an Iranian citizen
If USA saw your wife as Iranian they would not have issued her the ESTA. Because USA see her as German, and she has an ESTA, she does not need a visa, though as mentioned in a Comment by @Aganju, entry is determined by CBP at the point where entry is sought and, though largely theoretical, it might be refused, ESTA or visa not withstanding.
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