The 3rd level ability allows a Beast Master Ranger to "Choose a beast that is no larger than medium and has a challenge rating of 1/4 or lower" and "If the beast dies, you can obtain another one ..."
"Another one" means the same type of beast as the first i.e. no larger than Medium with CR 1/4 or less.
As an aside: the Beast Master Ranger is a relatively low-power class option; particularly if the ranger has no beast companion! Your DM should ensure that you can replace the beast as quickly and as easily as possible - right now you are about as much use as a Wizard without spellbooks, a blind fighter or an amputee rogue.
The Ranger's companion is ...
... is trained to
fight alongside you.
It is not trained as a mount. And ...
The beast obeys your commands as best as it can. It
takes its turn on your initiative, though it doesn’t take an
action unless you command it to. 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, Dodge,
or Help action.
The rules are quite specific here; you do not need to use your action to command your companion to Cast a Spell, Hide, Ready, Search or Use an Object. Now some of these are not very useful but Ready has definite possibilities - "Ready to attack the first creature that attacks me" for example.
Moving on to what a mount can do.
While you're mounted, you have two options. You
can either control the mount or allow it to act independently. Intelligent creatures, such as dragons,
act independently.
You can control a mount only if it has been trained
to accept a rider. Domesticated horses, donkeys, and
similar creatures are assumed to have such training.
If your companion is intelligent (unlikely) it will always act independently. If it has not been trained to take a rider (likely) it will always act independently. If it acts independently it is subject to the rules of the companion. If you command it, it will cost you your action is on the companion list of actions. If you don't command it, it acts as the DM sees fit.
If it is both unintelligent and trained to take a rider (like a horse) it must follow the rules for both a mount and a companion. If it is controlled, it can Dash, Disengage and Dodge at the cost of your action
The three options that any other mount gets for free (Dash, Disengage and Dodge) cost your action. So a companion horse is worse than a regular horse. Now, feel free to stop and call "stupid" but that would be a house rule.
Best Answer
Rules As Written, A beast companion can only activate a magic item during combat if you are incapacitated and it may require them to have focused on the item for an hour at some earlier point in time.
First some collected evidence:
From the question "Can a familiar attune and wear or use a magic item?" we learn that there are no rules forbidding any creatures from wearing, using, or attuning to any magic items which is also corroborated in the question "Can beasts benefit from magic items?" where it is stated that:
Additionally the question "Can an animal attune to a magic item?" explains that any animal (and so certainly beast companions) can attune to a magic item so long as they focus on the item for an hour. Upon attuning to the item:
The section on "Identifying a Magic Item" also states:
The fact that you can attune to an item even if it doesn't require attunement is explained in the question "Does a wand user know how many charges remain?"
Thus a beast companion will know how to activate a magic item and has the capability to do so if it focuses on the item for an hour. Whether or not you can convince it to do this will be up to your GM. That said, the beast companion's description states:
That list of actions comprises the only set of actions you can command it to take during combat. And if you don't issue a command, it simply takes the Dodge action. Thus in-combat you could not command the beast companion to use a magic item.
That said, the beast companion section goes on to state:
In this case, the companion acts entirely on its own, and thus is not restricted by the command rules. This would allow it to take other actions, such as Hide, or Use and Object, or using any magic object.
I agree that this is nearly nonsensical that a beast companion cannot Hide nor Use an Object during combat unless its master in Incapacitated, but that does seem to be what the rules state. I've now asked whether this is the case in the question "Does a Beastmaster Ranger's animal companion gain more actions when its master becomes incapacitated?".