I'm naive on the law around this, but it seems like Amazon.com shouldn't have standing here. What Perplexity does, from my perspective, is essentially the same as when I allow Firefox, Chrome, or Safari (or any other browser software) to see my credentials and access Amazon's website on my behalf.
Amazon certainly has standing for how YOU access amazon, even if it was in a regular browser.
Your feeling turned it to be correct. Amazon lost the case in the appellate court.
You're not using "standing" correctly, you're using it more to argue about the merits of the case. Standing is about filtering out frivolous lawsuits. One has "standing" to file a lawsuit when there is reason to believe these three questions could be answered in the affirmative, if a trial were to take place: Was the plaintiff harmed? Did the defendant cause that harm? Can the court rectify that harm?
It's not about what the answers to those questions are. Those are questions about the merits of the case, to be answered at trial. It's about if it should even go to trial.