I'm not sure if I understand your question. As long as you are signed into your Adobe applications and/or the Adobe Stock site with your Adobe ID there would be no difference on what you will see in your libraries on the two computers.
Could you give me some more information about the problem you are trying to solve?
Make sure you are using the latest version of the Adobe App you are using. About a month ago there was a small bug that has since been fixed. If your image in your library is indeed licensed it should not have a watermark on it in the library panel.
Basically why in my Adobe Stock library is the licensed photo showing a watermark? I can only remove the watermark by downloading to my desktop.
There was an update that fixes that. Update your apps to the latest version.