If all hardware and software is the same, you should get the same print. Which doesn't make the fingerprint globally unique, but unique enough to make you "one

If all hardware and software is the same, you should get the same print. Which doesn't make the fingerprint globally unique, but unique enough to make you "one of these 10000 people" instead of "any of this billion people". Combine it with other data, and the combined set identifies you uniquely.