Logic Pro: Same Sound Library on 2 Computers

Try this.

On the M1 mac,  rename the current "Library" folder on external drive to something else like "Temp_Library" (Logic won't grey out the drive after that). After that close Logic app and disconnect the external drive. Relaunch Logic and let it download and install Essential Sound library to system drive. This will wipe symbolic links created during previous library relocation. When Essential Sounds library is installed, connect the external drive and perform the relocation (just Essential Sounds will be relocated, which is needed to restore the symbolic links). Quit Logic. Now, delete the library folder containing just the Essential Sounds and rename the folder containing full library from "Temp_Library" back to "Library". Restart Logic Pro and it should now be working with the external drive library.

Then installing Logic Pro with all the additional media on both should give you identical libraries, and no need to use a hard drive to move anything.

Third party stuff registers itself to a machine, not a user, so you'll have to install them all again manually, and have them registered to the new machine, also. Moving the same library back and forth, I'd not attempt it, a lot can go wrong.