As in a number of earlier threads, there is a confusion between two separate things: what you are obliged to do by law and whether you will be checked that you comply with the law. This is a quote from the horse’s mouth (EU Commission webpage) but it ought to be common knowledge:
If you are an EU national, you do not need to show your national ID card or passport when you are travelling from one border-free Schengen EU country to another.
Even if you don't need a passport for border checks within the Schengen area, it is still always highly recommended to take a passport or ID card with you, so you can prove your identity if needed (if stopped by police, boarding a plane, etc.). Schengen EU countries have the possibility of adopting national rules obliging you to hold or carry papers and documents when you are present on their territory.
Driving licences, post, bank or tax cards are not accepted as valid travel documents or proof of identity.
Austria and Germany require this. You
can get fined up to €2500 if you are found out and the circumstances justify a fine. Italy
can refuse entry at the border. I did not check Spanish law.
As to Spain and albergues and their register of travellers: you can check this in the corresponding website of the Guardia Civil. Spanish nationals can use their driving licence as their ID, foreign residents in Spain can use their residence permit card, all others must show valid official ID such as passport or ID card issued for EU citizens by their country of nationality. I would not be surprised to learn that not every hospitalero/a is even aware of this. Let alone complies.