Should be possible, depending on how you actually set the image.
Simply Solid (F-Droid … sets just a colour instead of a picture as background), for instance, will ask whether the chosen colour should apply to the home screen, the lock screen, or both.
Since I suspect it generates a picture with the colour, because current Android seems to accept only pictures, I would suspect this home screen/lock screen/both choice should be there for pictures, too.
I’m not running the stock OS, and I don’t bother with pictures as backgrounds, so I can’t tell exactly whether there’s a standard way to do this, or whether this would need a different App or something else.
On thé stick version, press long enough the screen until toi can set a wallpaper. Choose one and then set if you want it only for the lockscreen, only for homescreen or for both.
Oh, weird. I am choosing an image from “live wallpapers” from an app , but that only provides the “Set Wallpaper” after choosing the image, and no options for the lock screen. If I choose one of the normal wallpapers, I get the lock screen options.
Now with a different wallpaper for the lock screen, I can go to the app and choose the wallpaper image and it now gives me an option to replace either the home screen or both the home screen and the lock screen.
OK, great. So I can choose just “the home screen” and it leaves the old image for the lock screen as it is.
Try the Google app “Wallpapers” from the Play store! There you can still chose between “Home screen” & “Lock screen” - a feature that seems to have been dropped from stock Android.