cn you have two system photo libraries on different computers with one Apple ID?

Is the external drive correctly set up for using with a photos library?

It must be formatted APFS or MacOS extended (journaled) (preferred for spinning drives)

It Must not be (or have been) used for time machine.

It must have the 'ignore ownership" checkbox ticked.

It should not be on a memory stick, SD or similar.

It must also not be a network drive (a direct connection is required - eg USB or Thunderbolt).

File sharing services such as iCloud Drive, Drop box, Onedrive etc are not compatible with Photos libraries.

It stands to reason that if you want any sort of reasonable performance it should be a fast drive with a fast connection.

See

https://support.apple.com/en-gb/HT201517

It depends on what you are trying to do. But the answer is yes

If you are syncing everything with iCloud photos - then the two libraries will be synced and will contain all the same photos.

If you are not useing iCloud photos, then the two libraries will be independent just showing the pictures importing into that library.

You could even have one of the libraries synced with iCloud (and your other devices) and the one on the other computer not synced and therefore independent.

Hi

Have you checked that the photos not on the second machine are on www.icloud.com

(Log in with your apple ID and go to the photos section)

Does the second mac have plenty of free disk space (enough for the complete library and then some)?

Yes, you can. It works fine.