This is probably the biggest thing I HATE about the companion app. If you log out for one reason or another, Home Assistant sees it AS A NEW DEVICE and you have to reconfigure all of the sensors over again.
I was cleaning up my login history today and removed the phone login in error which is why I am writing this. Is there a possible workaround for this?
I can see why it is annoying and I do not have an answer for you.
But out of curiosity: why would you log out on the companion app? I never felt the need, and if I do feel the need to log in as another user I do so in the browser. So without intending to sound like s smart ass, avoiding the problem is the best I can think of. But maybe I miss why you’d need to.
Another thing: if the device were to be hard linked and persistent to the device, how would HA respond if another user is now linked to the same mobile device?
ps. Andoid phones can login as another user. Could that be a workaround?
I made a mistake logging it out. I had to recreate my widgets as well which was a pain but that is generally only the reason why. I suppose I could always just do a Copy Paste in to my phone’s clipboard from KDE to the app with a long term access code. But, this is one of those days where I was caught by surprise!
If the Home Assistant app is hard linked to the device, I would think you could always disable it from the device list in Home Assistant. 
I have this problem too, mostly on my partner’s phone (which is a pixel 6) … however, the really annoying part is that they’re NOT actually manually logging out. What happens is a never-ending cycle of having to hand over their phone to me (they don’t really use the app itself, but our HA instance uses the data from the sensors), I log them in and activate the sensors that need to be active and things are ok for a while. Some time passes and I eventually notice that their sensors are all “unavailable” and when I pull up the app on their phone it’s logged out again. The “Manage app if unused” setting is already set to false so I have no idea why this is happening. It doesn’t seem to happen on my phone (also a pixel 6), but I use the companion app frequently.
I figured out that if you give your device the same exact name as what it was before, it links it with no further adjustments. The Widgets on Android though I still had to reconfigure.
But like you, mine has been very stable in keeping me logged in as I use mine a lot I guess it’s the widgets. My wife and daughter never have any issues and they don’t usually go into it and never have to sign back in.
Yeah, for me, having to reactivate all the sensors in the companion app settings is the biggest pain point. (and the way it breaks our automations every time they all go unavailable but that I can probably work around by improving those automations to account for it)