Not at all baseless.

Cats and dogs have other sensor hairs and whiskers, beside the ones on their muzzles. They use them to gauge the space around them, particularly their heads. That mesh screen dragging across all those hairs probably doesn't feel good. Though we obviously can't ask the cat, we can try to imagine what it's like, by thinking of what a similar fabric would feel like, rubbing against a highly sensitive part of our bodies.

The dog is a dog, and might be enjoying the sensation, or is a little slower putting the two things together.