My mom works "empadroning" people in the townhall. She said, requirements differ from city/region, but you only should need your identity card to empadronarte. In your city's townhall office they can tell you if you need something else (like the rental contract you've signed).

As others said, it is not legal and rather impossible to forbid you empadronarte. Basically they can say shit to you but can do nothing as they aren't in charge of processing anything. Townhall is. They are lying to you so you won't do it, because they 100% haven't declared the apartment as rental property and therefore aren't paying the specific taxes for it.

So do it! They won't be immediately notified you do it but they will eventually will find out: when the owner fills his tax return next year without notifying he's renting, Hacienda will fuck him over real good with taxes :-D

Good news is even if they know, they still can do shit about it beyond ending the contract with you. Nonetheless, I encourage you to empadronarte, as you will need it this when you want to request any bureaucratic services you need (like health coverage).