@Stuart ********
They aren't talking about NIE as just the number on a piece of paper as it doesn't prove residency. What they mean is Certificado de ciudadano de la UE (commonly known as NIE Verde or just NIE). All EU citizens living in Spain for over 3 months are legally required to have one. Non-EU should have a TIE.
NIE Verde is issued the same day IF the conditions are met (working for a Spanish company OR self employed in Spain OR proof of funds+private health insurance). TIE takes longer.