I think the only statements saying that they don't train on their customer data is from their CEO, but unless they encode it in their tos, that doesn't really matter.
https://blog.adobe.com/en/publish/2024/06/06/clarification-a...
Do you think they just stream any photos you've opened with Photoshop to Adobe?
Even without customer data to train on, they do have a pretty large moat with their image library. And you are right, they are in a good position VS competitors who trained on data they don't have rights to. We'll have to see how things play out legally, but I wouldn't be surprised if it ends up that something like midjourney ends up in an untenable position. However, openAI has a huge amount of funds that could be redirected to fight a oegal/lobbying battle. While they aren't a direct Adobe competitor, their whole business revolves around using unliscenced data to train their models, so they have a pretty clear horse in this race.
Or they just leave it up and admit they didn't need the rights and their library being cleared of rights isn't really a value add.