Historically, Mongolia had royal families such as Ashina of the Göktürks, the Borjigin of the Mongols and the Choros clan of the Oirats, etc. Most Mongol rulers since the 13th century were descendants of Genghis Khan, or tried to trace their lineage to him.
The term Mongolian Royal Family refers to the family of Prince Demchugdongrub, the puppet ruler of Mengjiang, a part of Inner Mongolia controlled by the Japanese during the Second Sino-Japanese War.
References
Royal family of Mengjiang Wikipedia(Text) CC BY-SA