The comparison with the balloon is perhaps not entirely accurate since birds like you say, fly using lift from the wings. So they need to have some relative speed vs the air.
The point is that as long as the entire body of air the bird is flying through is moving uniformly and without acceleration, it's perceptually indistinguishable from calm air (except visually, and even that only when flying pretty low).
For rotating and turbulent air, which would both not be totally unheard of in a hurricane, this probably doesn't apply though.
This is probably like when you swim out at the beach, and back and find you are 20m away from where you started due to currents. But you didn’t feel it.
With dead reckoning you could probably figure out.
You can’t feel linear/unaccelerated motion, and biological organisms aren’t great at indirectly deriving it from acceleration and rotation over time the way inertial navigation systems do.
For rotating and turbulent air, which would both not be totally unheard of in a hurricane, this probably doesn't apply though.