No, you should need to register to the Sony store only. When you create a Sony account, it is also creating an Adobe account for you. That's why when you register to the Sony account it needs to be the same email address and password that you previously may have used to register your PC with ADE.
I have a Mac, and ADE doesn't recognize any Sony ereaders on a Mac.

However, I think I may have read in other posts that ADE for PC does not recognize the T1 like it does the older Sony models.