As jcaron pointed out, details vary from country to country. Some general remarks:
In some European countries, residents are required to register with the municipal authorities of their primary place of residence. But that's really a detail for the Expatriates site.
It becomes relevant to the Travel site when you consider that hotels would be an obvious loophole -- there are places where one can rent rooms for months or even years. For this reason, hotels are required to identify their guests from the first day and to pass that identification on in certain cases.
Very broadly, driver's licenses or social insurance cards are less likely to be accepted in Europe than in the US. There is a tradition of issuing national ID cards, so there is less need to use other documents for this purpose. But any ID which is sufficient to cross borders should be sufficient in a hotel as well. (Many people forget that Schengen isn't about free travel without documents; it is merely about travel without document checks at the border.)