Lennie's background is only slightly mentioned with Aunt Clara but two Claras (Aunt Clara and Clara that owns the whorehouse) are mentioned in the book which gives us in insight into what his background is.
And I think Lennie's Aunt Clara is the same Clara that owns the whorehouse.
Lennie could have been the son of one of Clara's girls. He received mice and velvet a lot because the house had lots of mice and velvet from "nice chairs". George came into contact with Lennie because he probably came there often so Clara entrusted Lennie to him. His prostitute mother was most likely on drugs so that is why Lennie was not normal.
Plus Steinbeck could have chosen another name for Lennie's Aunt or the owner of the whorehouse but he chose the same one for a reason just like how he never gave Curley's wife a name to show that she was just an object to Curley.