If you want to count someone that gave birth then abandon their child and returning the child's christmas presents when we was around, then I guess yes he has a mother. Why do you think the father has full physical and legal custody? How about trying to answer the question objectively?
I have read the whole thread and I am now going to give you objective advice (and you have already received alot of objective advice)
A guardianship is suitable in a situation where NEITHER of the child's parents are available...ie deceased, disappeared or unfit.
A standby guardianship is a type of guardianship that ONLY kicks in if both of the child's parents are gone. (and is not solid unless no one else challenges it , other parent, grandparents, aunts/uncles etc.)
A stepparent adoption makes the child your child just as legally as if you had given birth to him, but that is only possible once you have demonstrated a long term, solid marriage and may require the permission of the child's mother.
A POA (Power of Attorney) allows you to act on dad's behalf if dad is incapacitated or not readily available to make a decision that needs to be made.
You need to start out with a POA. That would be more than sufficient to allow you to care for the child if dad is hospitalized or away from home for business etc.
Eventually, dad might wish to give you a standby guardianship, but he should talk to the child's extended family first, make sure there would be no challenging, and make sure that his state allows for standby guardianships.