Can't change permissions on external drive

Is "Ignore ownership on this volume" available, and can you select it?

For a number of reasons, it's not a good idea to perform general file storage on a volume designated for Time Machine backups. All my TM drives became "read-only" coincident with a recent macOS upgrade or subsequent update, so what you're describing is normal.

This is normal:

If it's available it will be at the bottom of the Get Info window. It's not shown in my screenshot because that drive is dedicated exclusively to TM backups.

Time Machine expects to use all of the volume dedicated to it. Its capacity will eventually be reached, after which TM will begin to remove old and expired backups. If you also want to use that drive for general purpose storage the solution is to partition it, bearing in mind that a catastrophic failure may render all its volumes permanently gone. If the video files on that same volume are important to you remember that TM isn't backing them up. It can't do that.

This "Ignore ownership on this volume" option is only shown for volumes that are not system volumes or are not TimeMachine volumes. If present, it is right below the "Sharing & Permissions" section in the "Get Info" window.

Note that the TimeMachine backup volume is "read-only" only for everyone user, but "read&write" for system and wheel users. So that's why normal users can't modify a TM backup, but the system can write to it during backups, as the TM backup is running as system user.

I agree with John Galt about not using the TM backup volume for "other" stuff. Regretting that decision later on is a problem you don't want to have. Don't mess with something so important as your backups...when you need em, they might become one of the most important thing in your life...Basically, the TM backup process scans the backup volume to determine there is room for the amount of data it wants to backup. By putting your stuff on the same volume, you could potentially cause it problems if it runs out of room in the middle of a backup when it thought it had room when the backup started.

If you really want to make use of some of the space on the TM disk, create two partitions - one for TM and one for your stuff. At least this way, your usage won't affect the TM backup space remaining estimate. But even then, your backup volume will still fill up eventually. When that happens, TM removes old backups to make room for new ones, and that's when you might come to regret the decision to not give TM all the space that is possible...

Good luck...