Why momentum of phonons called quasimomentum? Is it due to the fact that wavenumber of vibrations confined in a first Brillouin zone, and therefore can not have an arbitrary value as "usual" momentum?
The momentum of phonons isn't called quasimomentum. The true momentum of phonons is 0. While the true momentum is a consequence of translational invariance, quasi- or crystal-momentum is a consequence of the permutational symmetry of the atoms in the underlying crystal. See Ashcroft and Mermin, specifically one of the appendices.