Backing up Photos to NAS (Network Attached Storage)

The title of your question mentions "Backing up Photos", but If I understand you correctly, you are not asking about a backup (creating a safety copy of your Photos Library, that is never used, unless in an emergency, when you have to recover lost or damaged photos), but you want to have the working copy of your Photos library on a NAS, to be shared on your local network.

As TD explained, sharing a Photos library on a local network is not supported. This is clearly stated by Apple - see:  Move your Photos library to save space on your Mac – Apple Support

"To prevent data loss, Apple doesn't recommend storing photo libraries on external storage devices like SD cards and USB flash drives, or drives that are shared on a network."

However, you can share your library easily in iCloud as an iCloud Photos Library between all your devices. Then the central copy will reside in iCloud, mirrored on all devices (Use iCloud Photos to store photos in iCloud - Apple Support). But you will still need to make regular backup copies on a local volume (for example with tIme Machine) to be able to recover damaged or accidentally deleted photos (Back up the library in Photos on Mac - Apple Support).

You can't put a Photos Library on a NAS. There are two reasons for this:

One: the disk format. You are essentially putting a Mac database on a Linux computer. The Library risks serious serious corruption and data loss.

Two: The networked connection. The Photos Library is not designed to sit on a network share and you risk very poor performance and dataloss if you put it such a location.

The Photos Library is not just a folder that contains your photos and video. It's a complex interlinked collection of files, caches, metadata and versions, all managed by a series of interlinked database files.

If you want to manage your photos and videos from your NAS you need to use a different application for the job.