The distribution of the two-phonon $\ensuremath{\gamma}$-vibrational collectivity in the rotating triaxial odd-$A$ nucleus $^{103}\mathrm{Nb}$, which is one of the three nuclides for which experimental data were reported recently, is calculated in the framework of the particle vibration coupling model based on the cranked shell model plus random phase approximation. This framework was previously utilized for analyses of the zero- and one-phonon bands in another mass region and is applied to the two-phonon band for the first time. In the present calculation, three sequences of two-phonon bands share collectivity almost equally at finite rotation whereas the $K=\ensuremath{\Omega}+4$ state is the purest at zero rotation.