Make LR recognize external drive.

Perhaps I overlooked, but one thing not mentioned above.

If MACOS Catalina, consider the following

Outside of LrC, in the MACOS bring up System Preferences, select Security and Privacy, select the Privacy tab, scroll down to Files and Folders, for Lightroom Classic, and while you are at it, for Photoshop, make sure all media types are checked, tyicaly Removable Volumes is not checked if the  user error.

OK, an update because I initially missed the fact that you want to keep using both computers. You need to move the catalog folder from the old Mac to the SSD. The default location of the catalog folder is your Pictures folder. Once you moved the catalog folder to the SSD, you need to tell Lightroom (just once) where the new location of the catalog is. The easiest way to do that is to double click the catalog file inside that folder. That's the file with the .lrcat extension. You need to do this only once, but you need to do this for each computer. From that moment onwards Lightroom will start with the catalog on the SSD, so all the edits you made on one computer will be available on the other computer too. Just make sure you connect the SSD before you start Lightroom.

I get the feeling that you created a new catalog, and so Lightroom does not see any photos you previous had. You should copy your old catalog to the new computer and use that. Read this: 

https://www.lightroomqueen.com/how-move-lightroom-to-new-computer/

Lightroom is a catalog app, not a browser. It will not show any drives that do not contain any photos that are imported. So in order to let Lightroom 'recognise' a drive, you simply need to import at least one photo from that drive.

Hi Raymond,

In the images below I'm showing Adobe Bridge but the same issue is for Lightroom. 

Go to your System Preferences and select Security and Privacy (found in the top section). Now click on the Privacy setting (last tab). Now click on the lock on the bottom left and provide your Administrator code. Now select the Full Disk Access option and click on the "+" symbol. This will bring up the standard Mac Open window and let you add any application you want or need to have full access. Look for Bridge, select it, it will now appear in that window and make sure it's checked. Next go to the Files and Folders option and do the same thing as above (although there's nothing to check, just add. Close everything down and see if that makes it all work.

Please let me know if this doesn't work or if it does work! ;>)