Yes, there are two Mass obligations, the Sunday obligation and the Christmas obligation. A person cannot fulfill both obligations with one Mass. In order to fulfill a Mass obligation, we must attend either on the day of the feast or on the evening preceding (Code of Canon Law 1248). Therefore, a Sunday morning Mass could not possibly qualify as satisfying the Mass obligation the following day, since it is in the morning.
In order to satisfy the Sunday obligation, a Catholic could attend the Saturday evening Mass or Mass at some time on Sunday.
In order to satisfy the Christmas obligation, a Catholic could attend Mass on a Sunday at 4 p.m. or later or on Christmas Day itself on Monday.
It does not matter what kind of Mass someone attends so long as he attends Mass on either the day of the obligation or the evening preceding and does so for each obligation. Therefore, if someone attended a Saturday evening Mass for the Sunday obligation and then attended Mass again on Sunday at 6 p.m., that would satisfy both obligations, even if the second Mass was a Sunday Mass and not a Christmas Mass. Likewise, if a Catholic attended a 6 p.m. Mass on Sunday and a 10 a.m. Mass on Christmas Day, that would satisfy both obligations, even if the Sunday evening Mass was a Christmas Mass as well.
No matter how we slice it, two obligations equals two Masses.