How to save a edited PDF opened from google drive with Adobe Acrobat Reader
When you open a file from Google Drive with Adobe Acrobat, it doesn't actually open it in Google Drive because that is another computer that you can't access.

The file that you open with Acrobat on Google Drive is actually downloaded to your device and is stored in temporary memory. When you save the changes you make, they are not applied to the actual file in Google Drive, but to temporary file in the memory, which gets deleted once you are done with it.

Share option is different. Due Android's sandboxing and user restrictions, Adobe Acrobat actually never even accesses your Google Drive storage. It tells Android that the user gave a command to share this file to Google Drive. Then Android takes the file in temporary memory that had its changed saved and loads it into your Google Drive storage. So your complaints are not originating from Adobe Acrobat but due the way AndroidOS works and its could-have-been-better calibration for Google Drive.