I have to say unfortunately it's not. If you look at the crown on the one that you gave to your mum and the one that is
Rondafaye's picture, they are not the same and if you go and look at the Cartier website particularly the Santos Demoiselle in gold bracelet, the crown is also hexagonal. Also my price list (and some Googling) says that a Santos Demoiselle is around $20,000+.
This is not going to sound good but I have to tell you what I think straight and you are allowed to kill me for it. I think you are paying too much a premium on a no-name watch. For $13,000, you can get a Patek Philippe, a Breguet watch!
So I am going to say this, if you really did pay $13,000 for it, you bought into a very bad deal. If I was in your dad's situation, unless your mum opposes it, I would get a refund immediately because there are no certificates for anything (not even sure of the diamonds' grade) nor evidence to prove authenticity - we still do not know what brand it is yet.
If you don't trust me, go to the watchnut forum:
Timezone.com and ask them about it.