Welcome to the forum
@Creeping Snail (I like that handle)
What you are experiencing is normal and has to do with security. Depending on the security measures a site has in place, many (but not all) sites that require a user name and login recognize that you are using a new device and/or new browser and will refuse to log you in until you verify that it is actually you logging in. It can cause havoc for a user if he has not set up a 2 step verification method
within his account settings for that site. Two step verification consists of an alternate email address and/or telephone number that can receive text messages.
Even if you have your browser data syncing between computers, it does not get around 2 step verification(2FA). Some sites might send a link to your registered email but most require the alternate email address. By far the best and easiest method of verifying your're you is a cell phone. The site sends a code to the phone, You enter that code when it asks for it on the signin screen, and away you go. You only have to do this once per device and/or browser.