At the current state of the Game you have the following options with your companions:
They fight at your side, and with the special class hunter you can also learn them commands through your skill tree.
You can ride them only with special classes. Every one of the four ground classes, has at least one special class, which can teach youthe skill to ride or summon them
In case of the Sorcerer, you can insert your Boss Cards into your Grimoire and summon them as Companions. You can even mutate them to another Boss monster. That seems to be kinda random though.
The Falconer can also use its Bird companion to get up into air, were he cant get hit by attacks for a short time
As you see, there are already some options. But you cant tame Pets with a Hunter. Personaly i find this very boring, because the hunter feels kinda lucklust this way