Is a back-up drive exclusive for time machine?

Yes, absolutely, no problem at all using a drive for both TM and other general storage. If you are talking about an older iMac running 10.13 like in your signature, one way to use an external hard drive for a TM backup and other general storage is to partition it into two separate logical volumes. These appear to the macOS as separate discs, though they both reside on the same physical hard drive. Depending on usage, your hard drive life expectancy may be roughly one day beyond the warranty expiration.

that operation will erase all data on the drive, don't perform that on an existing drive with data you intend to keep, but for a one-time initialization before you write important data to the drive.

You could create a sparsebundle image using Disk Utility. Similar to hosting a TimeMachine backup on a NAS or other formatted HD. This is an easy explanation---https://www.imore.com/how-use-time-machine-backup-your-mac-windows-shared-folder

I do this frequently.

With older systems you could partition a TM drive and use one partition for TM and the other for general storage. BUT the danger in that is if the drive fails there is no backup of the general storage partition. The TM partition will be the drives you're backing up so you haven't lost anything yet. You have on the other partition. It's recommended to use a drive only for TM.

FWIW, since big sur the TM disk is given read only permissions. if you are starting from scratch with big sur, only TM can write to the disk. you cannot add anything else to the disk.

I think you are asking if a Time Machine external HD can be used for other types of storage. If I am correct the answer is NO, please never do that! The reason why is simple, all hard drives die, yes even TM drives. So when that occurs (and it eventually will) you will have not only lost data but it's backup!!!!!