Why would you use vanilla extract on your dogs nose?

I have no idea - there's no medical reason for this. I would guess someone somewhere claimed this was a good "natural" or "herbal" remedy for something, although I will point out most vanilla extract is artificial and has no real vanilla in it.

This is also a pretty bad idea - the risk of inhaling the liquid and causing inflammation in the nose is pretty high. Also, dogs have very sensitive senses of smell - this is the equivalent of putting an air horn to your ear and hitting the button. This would cause a great deal of distress and irritation to your dog, and there's no reason to be mean to your dog like that.

Additional information:

It is not put on the nose, but some dog breeders have used a small dab of vanilla extract on a male dog's collar to keep him from getting too whiny and anxious from the scent of a nearby female who is in heat. They say it partially masks the scent of the female.