There is a spell that can summon familiars, and a beastmaster ranger gets a companion, but is it possible to get an animal companion that is not a familiar without being a ranger or multiclass ranger? Without the animal being a horse, and it being able to attack?
I'm playing either a L4 Drow (Champion) Fighter or a L1 Tiefling (Draconic Bloodline) Sorcerer.
Unfortunately, your options are limited and many will require DM approval that is outside RAW.
A deep dip (5 levels) into Paladin or Bard is the only way to get the 2nd level spell Find Steed - which is exactly what you're looking for here (I think).
However, there are magic items that do the Druid bit of talking/being friendly with animals - but that doesn't really give you the control for attacking that you're looking for. However, a combination of the item and your DM may be enough:
Also, remember that a Panther has an AC of 12 and 13 HP. That's not gonna stand up long. You'll likely need custom barding (if you can even get it to wear it!) to boost that. But any natural panther (if your DM let you find/train one), won't last long.
The Bag of Tricks may be your best bet. Maybe your DM will change the animals or make it just a Bag of Panthers.
Druids have some abilities through spells such as Animal Friendship. Other than that, I think all you can do is buy or capture an animal and try to train it like anyone else. Just like in real life, dogs and horses are most amenable.
It would largely depend on your DM. For instance, on WotC's Dice, Camera, Action show, the party befriended an owlbear that helped them out in some combat encounters. However, I don't think there is anything in the official rules as written that specifically allows (or disallows) this.
5th edition is pretty easy to homebrew, and has a number of resources on how to do it well. If you want the ranger's animal companion without being a ranger, it should be possible to build it into another class in place of whatever kit that class gets. It won't come cheap, and it'll require a bit of extra effort to make the thing work, but it's certainly doable.
