We discuss Kondo behavior of a conduction electron system coupled with local optical phonon by analyzing the Anderson-Holstein model with the use of a numerical renormalization group (NRG) method. There appear three typical regions due to the balance between Coulomb interaction Uee and phonon-mediated attraction Uph. For Uee>Uph, we observe the standard Kondo effect concerning spin degree of freedom. Since the Coulomb interaction is effectively reduced as Uee-Uph, the Kondo temperature TK is increased when Uph is increased. On the other hand, for Uee<Uph, there occurs the Kondo effect concerning charge degree of freedom, since vacant and double occupied states play roles of pseudo-spins. Note that in this case, TK is decreased with the increase of Uph. Namely, TK should be maximized for Uee≈ Uph. Then, we analyze in detail the Kondo behavior at Uee=Uph, which is found to be explained by the polaron Anderson model with reduced hybridization of polaron and residual repulsive interaction among polarons. By comparing the NRG results of the polaron Anderson model with those of the original Anderson-Holstein model, we clarify the Kondo behavior in the competing region of Uee≈ Uph.