My question is the following, when we search for the bound states a finite well potential we have solutions symmetric and antisymmetric so we get two families of solutions. In this case, the fundamental state is the one with lower energy between the two families or each family has its own fundamental state?
A given potential has one ground state, even if this potential is symmetric. Nevermind the finite well, think of the harmonic oscillator, which is also symmetric: there is no separate ground state for the even or odd parity states, there is only one ground state. There is a lowest energy state with odd parity, but this state will have one node whereas the ground state will usually have no nodes, as discussed in this question.