Lemongrass isn't really a grass. It's an herb, both culinary and medicinal. We use it for landscaping and for really good hot tea.
Domesticated dogs are known to eat grass for all sorts of reasons. Dogs in the wild eat fruits and berries. Our dog at False Bluff gnaws on lemongrass for some reason including maybe that he just likes the taste. I don't think he swallows it. Lemongrass is really fibrous and it's difficult to eat. I've tried.