Amazon is a party to interactions with their website. That gives them standing to sue over those interactions. Whether they can sue Perplexity specifically is what was being debated in these motions.
Nah, once you have the bits on your device you should be free to do with them whatever you want. Amazon doesn't get to dictate how you consume what they give out for free.
I think I get it. Like they'd have standing to sue if I tried to hack their website, or change the price of items or something.