I never used to have this problem in the days before two-step verification. My tech-guru husband is loath to install our email addys on the laptop because of mail ending up on the wrong machine (he says. I don't understand it.)
You might be using SMTP instead of IMAP. SMTP is generally setup to remove mail from the server once the eMail client retrieves it. In such a case, when using two computers with the same eMail, whichever computer connects first is going to be the one to get the mail. SMTP can be set to save mail on the server, but a lot of people prefer not to leave things sitting on servers for security purposes.
With IMAP, the mail is stored on the server so multiple computers can access it for the same eMail address. But, then you have the aforementioned security concerns.
I can certainly understand not wanting to keep eMail stored on a server.
One way to get around it is to setup a forwarding address. Some mail servers will allow an eMail address to serve as both a regular and forwarding address. Let's say your eMail is me@myemail.dom. You would set that to also forward to also_me@myemail.dom. You keep me@myemail.dom for your desktop machine and use also_me@myemail.dom on your laptop. Then, when someone sends to me@myemail.dom, it goes to both your desktop machine AND your laptop.
That would be one way to resolve the Amazon authentication problem.
