Spatio-temporal scaling dynamics connected to non-thermal fixed points has been suggested as a universal framework to describe the relaxation of isolated far-from-equilibrium systems. Experimental studies in weakly-interacting cold atom systems have found scaling dynamics connected to specific attractors. In our experiments, we study a quantum gas of strongly interacting $^6$Li$_2$ Feshbach molecules, brought far out of equilibrium by imprinting a white-noise phase profile. The observed relaxation follows the same universal dynamics as for the previously observed formation of the order parameter in a shock-cooled gas of weakly interacting $^{87}$Rb atoms. Our results point to a single universal fixed point with a large basin of attraction governing the relaxation of quasi-1D bosonic systems, independent of their specific initial conditions and microscopic details.