Is it legitimate or an innovation for a person to title himself with a name other than his real name—such as "Abu Abd al-Rahman" for someone who has no children, or "Umm Yasir" for a girl? And is there any religious evidence for that?

It has been established that the Prophet, peace and blessings be upon him, gave a kunya (patronymic honorific) to those men and women who had no children. In Sunan Abi Dawud, it is narrated that Aisha, may Allah be pleased with her, took the kunya from her nephew Abdullah ibn Al-Zubair. And in Sahih Al-Bukhari, it is narrated from Anas that the Prophet, peace and blessings be upon him, gave a kunya to a young unweaned brother of his, saying: "O Abu Umayr, what did the little bird do?"