Every time we have gone through some form of transaction in Cyprus they have asked for our passport number. We gave our alien visa's instead recently when buying a car but no that wasn't good enough. When you draw money out of the account they need the passport - anything!!
In the UK the passports never came out other than for going abroad at airports. We never had the need for ID. What is the difference and why? I don't believe it's because we're not Cypriot as other people need ID as well.