Creative Cloud Enterprise may be offered in different ways and installed in different ways. The way we install it here on our college computers is with the licensing embedded and the use of most of the software (that doesn't need to connect to the Internet) is run in anonymous mode (no AdobeID login needed). Students here use Photoshop where it is installed without having to log in with an AdobeID. Some features that are only available through the Internet, an AdobeID is required (like for Muse) or other features internally in other programs within Photoshop, etc.
Most colleges that have licensed Creative Cloud Enterprise for their college computers have made availble to Faculty & Staff for their personal computers a discounted Creative Cloud plan. However, our college does not have the plan extend to Students ( an error in an email was sent out but later corrected) and I don't see that any other colleges have either purchased Student Creative Cloud plans but if they have they can chime in.
If you are having issues with your Photoshop trial on your pesonal computer, I would try to contact Adobe and find out why it is expiring before the end of the trial if that is what is happening.