I also get this same problem occasionally, but only on my PC which tells me I am signed in on "John's PC" and "John's Mac" and that I have reached the 2 computer limit, despite being on "John's PC". You seem to have got two instances of the same Windows machine.
I've tried to get Adobe to look at this, but it seems to slip between the Lightroom and the Adobe's CC authorisation teams. I think it's the latter, but I've not been able to make progress.
Uninstalling or cleaning the Adobe CC installation does no good.
So just sign out of both devices, and as you know you can continue until the next time.
I have exactly the same problem. A PC with LR Classic installed, and a laptop with the permitted installation. For some reason, each time I re-start my PC, Adobe forgets that I'm using my home workstation, requires me to sign in, and then to sign out for the PC that I'm working on! It seems that somewhere Adobe isn't able to log that I'm using the same machine for each new session. I spent HOURS with adobe support trying varous fixes, but nothing's worked so far. It's an annoyance! If anyone gets a fix, I'd love to hear...
One big issue that occurs if you sign out is as follows:
deactivates apps on your computer
Now, if you are signing out, then the question is why?
Note that lack of Internet connectivity, if you were previously signed in, and did not sign out, will not effect the Adobe Creative Cloud Desktop App in terms of it thinking you are signed in.
If the Adobe Creative Cloud Desktop App is getting signed out without your action, then something is wrong.
Some old users have a habit of signing out of the LrC, the individual app. This may come from way way back when. It is not required. It is not advisable.
The images you seem to be attaching are not showing in this thread. Don't try and attach images by email. Use a browser to log in to the forum, and then use the image 'widget' in the posting box to attach images/screenshots.
Sounds like you need to bring up Adobe.com, and inspect what devices you have activated.
Following is not the only way to accomplish that.
1. Go to adobe.com. You may or may not be signed in, sign in if you are not:
2. Click on your account button
3. Select manage account
4. Under the Plans and payments tab, select Activated devices
5. Look at what is activated, perhaps one is old and no longer used. Deactivate any as you see fit
6. Note that when you launch the Adobe Creative Cloud Desktop App, on your computer, and sign in, your computer should get added to that activation list. (as long as the 2 computer limit is not hit)
Followup
An alternative to starting at adobe.com, is to bring up your copy of the Adobe Creative Cloud desktop App, and click on Manage Account:
AdChoices