Does a controlled mount share its rider’s turn?

From “Controlling a Mount” on p.198 of the Player’s Handbook:

The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.

Does this mean that the rider and mount share a single turn?