This has got to be a common problem.
I have a personal Photohop/Illustrator account that I use at work for work, and at home for fun.
While at work I also need access to Acrobat Pro to edit PDFs which is provided by my employer. It's a real gargantuan pain to have to completely close Photoshop to work with Acrobat and vice versa.
Please fix this. As the Acrobat situation is likely the most common app this happens with, maybe just start with allowing that application to login to Adobe indepentent of the Creative Cloud Desktop.