It seems to me that Claudia the biological mother clearly favored Jacob, albeit for a very short time. After she makes the comment about having only one name she says to the woman "let me see HIM". She does not say them. I assume that during her pregnancy she was dreaming of her baby, what she would call him, what he would be like. Since Jacob came first he became that dream. The second baby was a surprise and thus Claudia had no time to develop an attachment to him.
Then the mother that they knew seemed to openly prefer MIB. She left him the game at the beach and seemed to see more of herself in him. I assume that Jacob was right when he said that mother was hoping that she would be able to hand the protector spot over to him. This can be backed up when she told BIB (boy in black?!?) that he would never have to worry about death.
Jacob was favored by Claudia. Then later on MIB wants to leave at the suggestion of Claudia. Jacob, I assume because he always felt second rate to MIB, stays with their "mother". Jacob wants and needs mother's love in some way that MIB did not. I haven't quite worked out what I think this means as far as their current day characters but I think it is an interesting look into the psyche's of the two. Jacob needed to be loved and needed. MIB was more interested in being independent. Jacob questioned his goodness while MIB seemed to be more confident in himself. Again, this is only half formed and I'm not quite sure where I'm going with it. Apologies.