Enríquez was born in San Juan to a poor family. The actual year of his birth is not clear due to contradictory dates, but the dates of 1674, 1676, 1680 and 1681 are referenced or recorded in official documents. Most of these variations were provided by Enríquez himself, who would report a younger age when questioned. Of those proposed, 1674 seems more likely. He was born to Graciana Enríquez, a freed mulatto slave who had inherited the surname "Enríquez" from her former slaver, Leonor. His maternal grandmother was born in Africa, with Angola and Guinea being mentioned while his maternal grandfather was an unknown white man. The name of Enríquez's father is not mentioned in any documentation, with the possible reasons for this being various and unexplored in the surviving records. It is possible that either the identity was truly unknown by the public or that the father was a member of the Catholic clergy, which would have prompted a "silence pact" to avoid a scandal. The second theory is supported by the fact that a member of the elite class, Luis de Salinas, served as the godfather of his brother, José Enríquez, despite the fact that he was also considered an illegitimate child. It seems likely that both Miguel and José shared the same father. Enríquez also possessed several sacramental objects and books written in Latin, which was a language only used by the clergy, now considered to have been inherited from his father. He was the youngest of four siblings, the others being María and Juan. José died soon after his birth, before reaching his first year. They lived in the room of a house belonging to Ana del Rincón on San Cristóbal y la Carnicería Street.