Relations

I have been living with a French family for the last 7 months. I eat with them, i help out with house work, i argue at times and i can even say whats on my mind. They treat me just like one of their kids. I am pampered and scolded, i am praised when i deserve and taught what is right and wrong when the time calls for it. I feel as much a part of this family, as i feel back home with mine. But in reality, i have no blood relation with any of the family members, and i am just someone who needed a place to stay and they let me stay with them all this while.
I never address them by name, i don't call them papa or maman, as their kids call them. I never call them by their names simply because they are more than double my age and the idea of adressing them by their names is just very strange. When i refer to them while talking to friends, i say my host parents. I sometimes call them my French maman or French dad too, but that seems very strange as well. It is true, my host mom has been like a mom to me. In the last 7 months, she has treated me just like her own family. I go to her when i have some problem, i talk to her if im feeling low. If i need some help, i turn to her, when i have questions its always her. She protects me and pampers me, she takes so much care of me. She will remember small things about my food habits, things i like to eat, and like any mother will make sure i have surprises and things i like always stocked up in the fridge or in my room. We might share a mother - daughter relationship, but i never call her mom, because in reality she is not my mom. It is weird, i have always started my sentence when i talk to her with "tu". It has always been, "tu veux un cafe ou un the ?" or " tu fais quoi ?". Never have i refered to her by name.
Today she was signing on some books she gifted me, and that is when we realized, i never call her by any name which she can sign by. It is strange, these relations that we make along the way of our life, which have no links, no basis, no history behind. They are just ties we make along the way which cannot really be given any names.