We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our results are inspired by, and partially extends, those of Anderson--Chen--Tarasca in the case of ordinary Grassmannian degeneracy loci to isotropic and odd orthogonal Grassmannians as well as maximal even orthogonal Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties.