List of things observed:

1. Rat stays on the mob, but doesn't hurt it.
2. Rat instantly falls off the mob and teleports to inventory
3. Rat attacks like normal, but eventually falls off.(this might be intentional, though I hope not)

I have noticed that these events are less likly to occur if you are midair while you throw the rat(I think it has something to do with hitboxes and the rat's "angle of entry"

Also, if you pick up a rat that is on a mob, it turns into an adult brown rat with no plague effects(it keeps being tamed though)