To Supplement @Pat Ludwig's answer, I would say that in that case each skill might give you different information about the monster.
To continue with the Dracolich example, a Nature check might get you that what you are seeing looks like the skeleton of a rather large dragon, and it shouldn't be moving around like that. Whereas a Religion check might give you that it is the undead remains of a dragon, animated by something vile.
What this means in terms of giving away it's abilities, you'd have to get creative. So your Nature check might give away that, even though it is undead, it has a fly speed, and that its bite is going to hurt, while the Religion check will tell you that it has typical undead traits, and probably has Resist Necrotic, and Weakness Radiant, or something similar (I don't actually know the stats of a Dracolich offhand).