Diluted magnetic semiconductors (DMSs), also referred to as semimagnetic semiconductors, are a class of semiconductor alloys whose lattice is partially composed of substitutional magnetic atoms, such as Mn, Fe, or Co. One such effect is that charge carriers, bound to a donor impurity in DMS structures, can polarize the spins of the magnetic ions within their vicinity. This complex, which consists of a charge carrier bound to an ionized impurity and surrounded by magnetic ions with locally aligned spins, is referred to as a bound magnetic polaron (BMP). In this study, we used computational methods to analyze how the central barrier and manganese composition affect the electronic properties of a magnetic impurity confined in a double quantum well composed of a diluted magnetic semiconductor, Cd_1-xwMn_xwTe/Cd_1-xbMn_xbTe. We also employed the effective mass approximation and variational technique for numerical calculations. To compute the spin polaronic shift, we used mean field theory with the modified Brillouin function. The main findings of our work reveal that an increase in barrier thickness enhances the binding energy of the ground state of an impurity located at z_i = (L_w+L_b)/2 for all manganese compositions. Moreover, a higher composition raises the height of the barrier potential, where the effect of quantum confinement is very strong, leading to a rise in the magnetic impurity binding energy. On the other hand, the spin polaronic shift follows the same trend as the binding energy with respect to the aforementioned effects. Finally, the exchange interaction between the magnetic ion moments and the spins of conduction electrons in DMSs has paved the way for advancements in spintronic device technologies. We anticipate that this study will offer valuable insights into the electronic properties of double quantum well made from diluted magnetic semiconductors, which could prove beneficial for spintronic applications.