Moving Photos Library

The normal user data migration process should include the Photos library, including the folder/album organization. If for some reason that does not happen, dragging the library as a unit would include the folder/album organization.

I agree with Mark, just a little add on.

When you set up the new Mac and transfer the documents and dato from your new Mac, do it immediately, when the new Mac asks you to transfer the documents and data from your old mac. Connect the old mac or your Time Machine volume do transfer the data immediately using the Setup Assistant. It is tempting to finish setting up the Mac and then to use the Migration Assistant later. Do not fall for the message "You can set up the mac later with Migration Assistant". This is very problematic, when you have already created a new user account. The migrated Photos Library may have problems with the file ownership and permissions. Use the "Setup Assistant" at the first prompt, before you create a new user account.

I don't know how much code they share, but they occur at different times. Set-Up Assistant happens during initial setup for the Mac, before there are any users created. Migration Assistant (MA) can be invoked later, but it has less of a clean slate to deal with. As noted above, this can have permissions effects, as some uid codes may be assigned before it runs.