FWIW, people are already using Apple Pay and Google Pay through a phone with fingerprint sensors. So... it's really not a new thing we're sacrificing privacy for convenience.
I use biometrics locally with my phone, but do not transmit that data to Apple, to Amazon, to MasterCard, to anyone. All they get is the usual credit card info.
Amazon it trying to cut out the middleman here, but the middleman they're trying to remove is under my control, and Amazon definitely isn't, so the shift is enormous.
This is your entire hand, and it's stored off-site and owned by a company who at any point in time can change their mind about that data and sell it to whoever will pay for it.