This is one of those cases where I think whoever wins, we all wind up losing.
If Amazon wins, there won't be any AI allowed there except those that Amazon has agreements with. Or Amazon's own AI, designed to encourage purchase of whatever Amazon wants to sell more of this week.
If Perplexity wins, shopping websites could wind up working like Ticketmaster and it's 'anti-scalping-bot proofing' that doesn't actually stop bots or scalpers.