Is it true that concealment (ستر) is a unique characteristic of Islam compared to other religions, and that sins before Islam were recorded on the sinner's door, such as writing "thief" on the door of one who steals? And if so, what is the proof for that?

The summary of the answer: This report has not been authentically attributed as a saying of the Prophet, peace be upon him. However, it has been authentically or well-attested as a saying of Ibn Mas'ud, may Allah be pleased with him, as well as some of the Tabi'un (successors). It is likely that this report is among what the early Muslims received from the Children of Israel.

Concealing the sins of wrongdoers is an attribute of Allah; He is the Concealer (Sattir) and loves to conceal the faults of His servants. This is not exclusive to this nation (the Muslim Ummah).

The difference between the two nations: For the Children of Israel, repentance required specific expiations that they had to perform. In contrast, the Sharia of Prophet Muhammad, peace be upon him, made the door of repentance easy. As soon as one resolves to abandon the sin and seeks forgiveness, the servant's repentance is accepted, as stated in Allah's words: "And whoever does evil or wrongs himself then seeks forgiveness of Allah will find Allah Forgiving and Merciful" (Quran 4:110). Expiations in Islamic Sharia are related to specific, limited matters, not to every sin.

And Allah knows best.