I also really hope there's a kind of transparency effect or something associated with this. Looks silly just being able to walk through them normally.

I'm guessing the idea behind the name is that cats stalk silently, and the only cats in Eorzea would be coeurls.

General FYI: the Japanese name for the ability (also tentative) is "Equip Stealth" where "Stealth" is the English word.