In every play I've done I've planted and harvested at least one growing season's worth of haygrass. I've taken the hay along on caravans and left it out for my herbivores. None will eat it. They'll munch on my kibble before they touch the hay.
What is the signal than an animal eats hay. I checked a couple of animals I know eat hay in real life. Here's their list of "foods".
vegetables
seeds
live plants
meals
processed foods
kibble
liquor
I don't see Hay, and they don't seem to see the hay waiting for them to eat it.
Every animal that eats live plants also eats hay, that doesnt mean they dont prefer other kinds of food though. Take the kibble away and they ll eat the hay.
All herbivores can eat hay, as well as some omnivores like monkeys. But generally it's only herbivores or "prey" animals. Dogs and bears don't eat hay, and neither do predator animals like cats, wolves or or wargs.