Well, running has a strong eccentric component for both the quads and calves. Walking up a steep hill and biking don't.
In running a significant amount of energy (up to 50% of total energy in some estimates) is stored in the achilles tendon, arch of the foot, and some other structures. It's much less in uphill walking and less still in biking.
Running has a short explosive foot strike. Walking uphill and biking have a long continuous force production.
Running uses no knee extension. Walking uphill uses some and biking uses a lot.
So in a number of ways it's closer to biking but definitely quite different from both.
I agree that hill running can be great for running as long as it still resembles running. I also try to use the treadmill for things I can't easily do outside. You have a treadmill that allows a 16 degree decline? The ones at my gym go to 5 at most.