Speaking as a naturalist, and with nothing whatever to back me up, I suspect it's just "eye-ness"...some fish have eyespots also, near the tail of course, presumably to help predators to aim wrong.
The eyespots in American moths and butterflies definitely antedate he doemstic cat.
There were plenty of wild cats in the Americas. Anyhow, generally speaking you're certainly right, and even more certainly right enough that we need not argue over it! The moths of course aren't actually aiming for any particular critter. The question is what the bird thinks it's seeing, and presumably the answer is simply whatever that bird fears might be hunting it. In some cases those are cats, in others they are owls, in yet others they are hawks, and so on and so forth.
The butterflies are just inadvertently homing in on whatever is keeping the birds away.
PS. And the reason why I wanted clarification is because it's one thing for a moth to mimic eyespots, it's quite another for a moth to mimic a bird's entire face. What I pictured from the original post was totally different than what seems to be the issue ("eyespot" scare patterns).