I'm not willing to test in (I don't buy from Amazon) but almost all the time when ordering online from Denmark, I have to authorize the transaction with the second factor — phone app or TOTP key etc.

This applies even when I'm a repeat customer.

It depends on the payment method, but for Amazon for me that means either using the Dutch Ideal system which means I have to use my bank's authenticator (which takes my debit card and asks for a PIN) or my bank's app (PIN plus unlocking my smartphone) to approve the transaction, or to get asked for my credit card's 3dsecure password and (often) a confirmation code sent by text (again requiring unlocking my smartphone).

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.

It is and I would love to know when exactly that is the case. Normally all my online payments require 2FA, but some companies can apparently just transfer my money automatically. I'm guessing they need some kind of agreement with the bank, as otherwise 2FA would be pointless.

Nobody is saying it isn't. Parent was saying the burden is on Amazon because they're allowing this to happen without further verification - so in the EU Amazon wouldn't be able to enforce a "you bought it, you're stuck with it".

It is, yeah. But it’s not legally binding if a child misuses that mechanism.

Yes it is, at least in France. I do it all the time with Amazon