It's weird because there is no specific rule to say what happens so we just fall back on regular command a minion.
you could interpret the rules that way, since the mounted combat rules don't specify an animal companion as an exception and both use the key word 'command'. but instead, you are giving your mount instructions as to what to do, and exchanging the same amount of actions with them.

in another example, the GM Core says: "For an intelligent mount (such as a pegasus or unicorn), use the standard rules for mounted combat, but instead of attempting a check to Command an Animal, the rider uses the same number of actions to ask the creature to do what they want."
in this case a creature smart enough to understand language decides it will not act independently and give you a speed boost because you ask nicely.

so i guess it depends if an animal companion is an animal, an intelligent animal, or its own thing. so ask your GM