The nature of spin freezing in geometrically frustrated icosahedral quasicrystals Tb-Mg-Zn and Tb-Mg-Cd was studied by thermoremanent dc magnetization (TRM) decay as a function of aging time and magnetic field. At low temperatures the magnetization exhibits typical broken-ergodicity phenomena, as characteristic of spin glasses (SGs). However, the observed linear dependence of the TRM on the magnetic field in the low-field regime is incompatible with the aging of a nonergodic system in an ultrametrically organized free energy of a SG, but compatible with a single-global-minimum free energy of a superparamagnet below the blocking temperature. The Tb-Mg-Zn(Cd) quasicrystals are, from this point of view, different from site-disordered SGs, but similar to geometrically frustrated pure (site-ordered) systems, like the kagomé and pyrochlore antiferromagnets, which also exhibit a superparamagnetic component in the magnetization below the spin freezing temperature and clustering of spins. The Tb-Mg-Zn(Cd) quasicrystals show features associated with both the site-disordered SGs and the superparamagnets. This duality is not a specific feature of spins in a quasiperiodic structure, but is found quite commonly in nonrandom (site-ordered) geometrically frustrated magnetic systems.