In this case, the verb "to register" means "to perceive" or "to understand" or "to recognize." Think about a cash register in a store. The clerk pushes numbers on the register and the register receives those numbers. It's the same way here. The baby is receiving/perceiving the other person "as a blurred self-extension."

A good synonym would be to recognise him as…… To register in this case means to acknowledge ……

It simplyeans that the baby's brain thinks the speaker is an extension of herself etc.