The water doesn’t stop right away when I turn off the UltraStream undersink faucet, is it faulty?

No, the reason the water doesn’t cease right away is because the faucet is a specially designed vented faucet.  This allows the UltraStream to breathe which is very important because of the molecular hydrogen it makes.
A vented faucet doesn’t turn the water off at the actual tap but turns the water off before it actually water goes into the UltraStream.

This means that if the UltraStream did break for any reason the only water that would leak out would be the water in the actual UltraStream and hoses.  This would be about 250 ml (a glass) of water.
With a vented faucet, as long as the tap is turned off on the sink top there will never be any leakage from the mains.