How sound is the Hadith that links sleeping postures (on the stomach, on the back, on the side) to belonging to specific peoples, and what is the wisdom behind that?

We do not know of a hadith with the specific wording mentioned, but authentic hadiths have been narrated that clarify the Prophetic guidance regarding sleep. The Prophet, peace and blessings be upon him, commanded lying on the right side and forbade sleeping on the stomach, saying: "Indeed, this is a posture that Allah dislikes," and in another narration: "This is indeed the posture of the people of the Hellfire."

In the Quran, there is an indication that sleep can be on the right and on the left. Ibn Al-Qayyim mentioned that the most beneficial sleep is sleeping on the right side for the food to settle in the stomach, then shifting slightly to the left side to hasten digestion, then settling into sleep on the right side.