2 PC's Running Directly off Single Router Question

[TRIALnERROR]

Maybe someone can shed some light on my situation.

I run multiple Amazon accounts. I purchased a new gaming PC today. I will be using this New Pc for personal use/gaming. I have my monitor hooked up to the New PC and the Old PC. I switch back and forth.

My question is when logging into personal accounts on my NEW PC which is connected into the same router as the old PC is still using and connected to, is there any way Amazon will link through the router to old closed Amazon accounts that match up with the Users and Passwords I will be using for personal use?

I will be using two completely separate computers. One for Amazon only, and the other for personal use only. Both are connected to the same router. Is this safe?

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is there any way Amazon will link through the router to old closed Amazon accounts that match up with the Users and Passwords I will be using for personal use?

Yes AZ can - this is not safe.

[TRIALnERROR]

Then how is it different from "Users" vs new computer? I have a personal user account with using logins from closed accounts, but I switch users to log into Amazon Ghost accounts.

If I turn off the "az computer" and then switch to my new computer using old logins, this should be safe right?