Photos library on external drive not opening

And the external drive needs to be compatible with Photos for Mac. prepare the external drive exactly as described here: Move your Photos library to save space on your Mac - Apple Support

The essential points are:

• The file system format needs to be either MacOS Extended (Journaled) or APFS
• The external drive must not have been used for Time Machine backups or have any system software or backup software installed
• The drive needs to be mounted locally with a wired connection, not over the local network.

Your new drive seems to be formatted correctly - the only thing you should also check, is if the formatting is case-sensitive. If yes, that could be causing problems. Make sure, the format is not case-sensitive, or you may encounter problems with videos.

You can tell Time Machine where it should put the restored library.

The user Guide does not mention it, but when you have opened Time Machine and select your Photos Library in the time tunnel, hold down the CTRL-key while you click the library, then use the commend "Restore to ..." from the contextual menu and save the restored version to your other external drive.

You say you copied your iTunes and Photos libraries from your Time Machine drive, but that may not have been the right approach. The correct procedure would be to perform a Time Machine "restore" from the T M drive to the target drive, which will be able to reconstruct the complex structure of the library. That is definitely true of the Photos library and may be true of iTunes/Music libraries.

Jimzushi wrote: … The external drive is formatted APFS GUID Partition map. I also unchecked "Ignore ownership on this volume"

APFS is good. But "ignore ownership" should be checked:

As to restoring from Time Machine, there is a version with pictures and explanation:

Time Machine Restores a Photos Library - Apple Community

This

Restore a photo library from Time Machine on Mac – Apple Support (UK)

has the details.

The picture shows the "Enter Time Machine" location. If it's not there, then this may not be a Time Machine volume. Time Machine data has to be created on the Time Machine drive-- you can't move functioning Data files from a Time Machine volume to some other volume. A Time Machine volume must be specially prepared, and nothing else can exist on the volume. Are you sure this is a Time Machine volume? This is exactly how a Time Machine volume should look:

As far as I know, a Time Machine volume can't be restored to another drive. If you have a proper Time Machine, you may have to make a bootable drive, boot your Studio from there, and restore that way. But we are now well beyond my experience!

See this: Use Time Machine on your Mac to back up to multiple disks – Apple Support (UK)

• Connect the backup disk that is holding the old backup to your new Mac.
• Make sure, the Time Machine icon is added to the main menu bar (in the Settings > Menubar > Time Machine).
• Click  in the menu bar, press the Option key, then choose "Browse Other Backup Disks". and select the drive with the backups of the Photos Library.

You need to connect the backup drive, where you initially copied the Photos Library from. Do you no longer have this drive? Or is it the same drive where you are now making your backups to?

I am getting mixed results, when I try to "Restore to ...", Richard. If the original folder exists, Time Machine will use the original location to restore it to. But if this folder does not exist, I am prompted to specify a different location to restore to (I tested on macOS 26).