changing the name of a disk volume

Are these macOS boot drives? If so, then changing the name is tricky since a macOS boot drive utilizes two separate APFS volumes...one is a read-only system volume and the second is the writable volume containing the home user folders. Unfortunately macOS won't properly rename these boot volumes. If you attempt to rename a macOS boot volume through the Finder, then it will only rename one of these two volumes.....yet these two volumes are linked together behind the scenes. The only way I have found to rename these macOS boot volumes is by using Disk Utility to rename each volume so they stay named as a matched set. I have never tried to rename the Volume Group instead. Instead of making things better with later releases of macOS, Apple seems to have made this worse since at one time renaming just of one these paired volumes would automatically rename the other to match when using Disk Utility....sadly this is no longer the case. I have not tried to rename the macOS system volume on macOS 11.x+ for a while now so I have no idea if it is even still possible.

I'm not sure this is what is happening here, but you did mention a backup drive without mentioning any details about the backup drive. These details are critical if you want help since there are too many unknowns after so much back & forth already.

I guess the only other possibility is are you running any anti-virus apps, cleaning/optimizer apps, or third party security software? If so, then maybe one of these types of apps are causing the problem.

Is this Mac being managed by a school or employer? If so, then perhaps they are not allowing such changes and you should contact them for assistance. If this Mac was purchased used, then it increases the chances of it still being managed by a previous owner.

Are these macOS boot drives? If so, then changing the name is tricky since a macOS boot drive utilizes two separate APFS volumes...one is a read-only system volume and the second is the writable volume containing the home user folders. Unfortunately macOS won't properly rename these boot volumes. If you attempt to rename a macOS boot volume through the Finder, then it will only rename one of these two volumes.....yet these two volumes are linked together behind the scenes. The only way I have found to rename these macOS boot volumes is by using Disk Utility to rename each volume so they stay named as a matched set. I have never tried to rename the Volume Group instead. Instead of making things better with later releases of macOS, Apple seems to have made this worse since at one time renaming just of one these paired volumes would automatically rename the other to match when using Disk Utility....sadly this is no longer the case. I have not tried to rename the macOS system volume on macOS 11.x+ for a while now so I have no idea if it is even still possible.

It isn't tricky.

It doesn't matter if the Data volume isn't renamed.

It doesn't cause any problems.

I rename all of my startup drives.

Apple Silicon seems to set the Data volume name to "Data."

My Startup drive is named "Amber" and my Data volume is named "Data"

Share what you know and get recognized - Apple Support

Perhaps if the user ( you ) were to give us a Screen Shot from within Disk Utilities we could better understand what you are talking about

Be sure to use the " View ALL Devices "

Have you restarted and retested?

If you mean Time Machine backups they can't be renamed. They are read only.

No, I mean DISK VOLUMES.

A Time Machine backup drive is a "DISK VOLUME." You can't rename it.

What is the format of the "DISK VOLUME?"

A screen shot of an icon imparts no additional information for anyone to try and help.

Everyone here understands you can't rename some disks. You need to include requested information so someone can figure out why. It does no good to keep repeating the same thing.

Good luck.