I had rust on my chives this year, is this the same species as the one that affects leeks?

Yes! the leek rust fungus can attack chives and garlic as well as leeks.

On chives the attack is often sufficient to kill off all the leaves and it is likely to be perennial within the plant, therefore the only satisfactory treatment would be to dig out and burn the plants.

If possible avoid planting out Leeks, Garlic or chives in the same area for a couple of seasons.