Is there a mechanic by which a non-ranger character can have a useful animal companion? The PHB has animals that you can buy, but how can a character make use of them if they are not a beastmaster, have the find familiar spell, or an animal friendship spell?
As an example, I would like to find a way to have a fighter use a hawk for help with survival, hunting and possibly even to harass an enemy from time-to-time.
Best Answer
No, at least not for combat
The find familiar spell and the beastmaster archetypes are significant class features. In particular, the beastmaster archetype advances with levels. On the other hand, the animal friendship spell doesn't give you control over an animal; it only charms it.
To give another class the ability to use animals in combat would be something akin to giving the fighter the paladin's lay on hands ability, or a wizard bardic inspiration.
Noncombat uses are probably ok by animal handling
If you want an animal for a more mundane, noncombat use (a pet dog, maybe?) you could probably get away with just using animal handling. Importantly, this does not mean that you get to control the animal like you control your character. It would act under the GM's discretion, with its obedience affected by how well you roll. Thus, it's less like the wizard's perfect telepathic synchrony with her familiar and more like a real-life animal handler with a trained dog or horse.