Mean-field theory for the ordering transition in the majority-vote model on multiplex networks with two independently generated layers in the form of scale-free networks is presented. In this model two-state agents (spins) located in the nodes update their states according to the opinions of the majorities of their neighbors in both layers with certain probability related to the degree of the internal noise. If the opinions of the majorities of the agent’s neighbors in both layers coincide the agent adjusts her opinion to them with higher probability; otherwise, she acts independently and makes the decision for or against randomly, with equal probability. An interesting property of the model is that within the mean-field approach the nodes with odd and even number of neighbors within each layer should be taken into account in a different way. The critical values of the internal noise evaluated theoretically for the layers with different degree distributions agree quantitatively with those obtained from Monte Carlo simulations.