Is the money paid by the German state to a resident for his living in his brother's apartment, then moving to another, cheaper apartment while remaining registered at his brother's address and continuing to receive the old amount, considered unlawful money (haram)?

You are not permitted to conceal the matter from them in order to take more than the actual rent you pay, as this is forbidden deception. Rather, you must inform them of your move to another dwelling and tell them its rent. If they agree to leave the matter as it is, there is no sin upon you in taking the excess; otherwise, it is not lawful for you to do so. Muslims residing in the West must be good role models in fulfilling agreements, for the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions."