Amazon do this because it makes it frictionless for the legitimate consumer - but at risk of letting through unauthorised transactions.
There is no way to just click 'buy' and have it delivered. Only AliExpress does that for me. Perhaps this is possible in the EU with Amazon Prime? All Dutch online shops use Ideal, so accidentally ordering something there is just not possible unless you give your kid access to your smartphone and PIN.
This applies even when I'm a repeat customer.
This comes with extra liability but this is part of their risk and costs.
Asking for a CC, password, fingerprint/passkey, authenticator, email confirmation?
If you’re claiming that no security is possible, might as well allow anyone to login on any account.
Alternatives DO exist.
The solution (that many people do not want to hear) is very simple: supervise your kid, be a parent.
Edit: I love getting down-voted for saying "be a parent". Laughable. We are on HN and you do not know how to prevent this from occurring? Read the other comments, there are lots of suggestions. You have absolutely no excuse.
Or let your kids use a different user account on the computer.