Gordon,I have a tom cat called Oscar,we also have 6 dogs that he gets along with just fine, but most of the time he prefers to keep to himself.
His people time is mostly with me while I shoot in my back yard,,and he does much the same thing your's does.
He lays on top of my target bale an watches the arrows come in,doesn't swipe at them but you keep see him following their flight in like he's about to ponce on them.
Turns him from a mean old tom cat into an instant kitten,,,,funny as heck,but freaks out visitors that see him up there while I shoot.
The Mrs just shakes her head an mutters "your both nuts",,,I'm ok with that ;^)