I still have some doubt about the rules anyway.
I saw many DM ruling it different.
For what i got till now, other interpretation of this is:
According to Ranger's Companion rules, "It takes its turn on your initiative. On your turn, you can verbally command the beast where to move (no action required by you). You can use your action to verbally command it to take the Attack, Dash, Disengage, or Help action. If you don’t issue a command, the beast takes the Dodge action."
As such, regardless of whether the companion is a controlled or uncontrolled mount
it is going to take its turn on your initiative, and in fact its turn will overlap with your own, since your action determines what action it takes or can take.
According to the Mounted Combat Rules, a controlled mount "moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge". So, if you controlled your mount, you could use your action to verbally command it to take the Dash or Disengage actions, and if you don't command it with your action, it takes the Dodge action. However, since the actual benefit of controlling your mount is that its turn partly merges with your own, and since your companion's turn already does this, there is no reason to control your Ranger Companion.
According to the Mounted Combat Rules, an uncontrolled mount bearing you "puts no restrictions on the actions the mount can take, and it moves and acts as it wishes." So, if you leave the companion independent, you actually still have all the benefits of a controlled mount by virtue of the Ranger Companion rules.
Said in few words:
- when you mount something you can choose to control it for it share your same initiative, but losing some of the mount possible actions
- The ranger animal companion already share your initiative, so have no sense to control it.
Any mount dont normally share initiative with their master, so the master need to control them for get it, but the ranger companion does it, so even uncontrolled (even not mounted) it will keep doing it. I dont know if this is correct but seem make more sense to me. Is it wrong?