Single H Bridge to control multiple DC motors
No, that won’t be suitable for at least three reasons.


The mux or shift register will not be able to handle motor currents
A H bridge is used to reverse motor direction and this means reversing polarity and I’ve never come across any semi complex silicon product that would survive something like this.
Back emf from the motor are handled by the H bridge rather nicely using diodes or MOSFETS with diodes but a mux or shift register will just probably die.