@derei thank you for raising a very good point.
Screen sharing is not an option. (1) Same security issues - someone could see the unattended device and since the browser is being used all the time, it could mean that (2) the device is open to much more abuse and (3) the process of connecting to a remote etc totally ruins the UX.
Also many people use large resolution screens on one main machine and then may wish to access via mobile or smaller desktop device. screen sharing from dekstop to mobile is hopeless. and screen sharing from large high resolution to smaller lower effective resolution is messy with window management.
This is how I would see it possiby working. Assume 3 devices, A, B, C. Assume device A is being actively used. Device B is in hibernate, locked or completely off and C is being used by someone else.
For device B (off, hibernate, locked), the browser open tabs and stack will only get updated when the user logs in and access the broswer. any tabs requiring login will remain at the login page as session ids would be different.
For device C (someone else using device), the browser would update, but it would be the primary users choice to allow this (same as when a browser is logged into on a public PC at the library for instance).
However, once again, for secure sites like banks etc, or sites requiring two factor auth such sessions will stop at the login stage on device C and go no further as the session ids will not be initiated. i.e. the act of entering login information would only work on device A. With screen sharing this security does not exist.
With device C, if the user has a "remember me" flag set on any site which requires only a simple login password, there is a possibility of security risk, however I believe that Vivaldi can use the sync functionality to block all such aiuto logins on secondary devices.