My sister's playing a paladin with a raven as a pet. A friend discussed having a dog for his hunter. I'm presuming you can use Animal Handling to train and control the animal. I'm also presuming you can train your animal to attack and follow simple commands.
So, the question is, why can't you do this with a ranger's animal companion or a wizard's familiar? From what I've seen, trying to control your animal as a ranger requires you to expend an action each round. If you have a familiar, you can't use it to attack unless you're a warlock with the Pact of the Chain.
So... does this mean it's actually better to have a pack of dogs you've trained for combat, and not have them as part of a class ability, because they're more useful? Or should my sister's paladin not pick up ritualist to turn her raven into a familiar, because it might actually be more effective as just a normal raven?
Alternatively, get an animal, and find a druid to cast Awaken on it, then let it get its own class levels over time.
It's reasonable to use the Wisdom (Animal Handling) check to train or control an animal, if and when the DM decides that the player-described action involving those goals has an uncertain outcome.
A ranger's animal companion is, generally speaking, better than a normal animal on which it is based. Further, no Wisdom (Animal Handling) check is required to control them and they get better as you level. So you're trading a risk (the ability check) for a cost (an action). A wizard's familiar can't attack, but can do other actions and has valuable traits none of which require an ability check to use. The cost is a spell slot and some gold each time the familiar is killed and brought back.
A pack of dogs trained for combat might be better than these class features, given certain circumstances, at certain levels. But any time your action is left at the mercy of an ability check, things can go wrong. They also don't get any better with level and take time to train that might be better spent saving the world from the villain-of-the-week.
Actually, I don't believe animals are prevented from getting better. The Monster Manual gives advice for augmenting creatures, so I'm perfectly fine with giving animals a boost over time as they get involved in conflicts and survive. A veteran wolf is going to probably be better than a whelp, and if the character takes good care, I'm sure it can live a good long time. (Throw magic in the mix, and it's anyone's guess). The Druid casting Awaken can give the animal an Intelligence of 10, which in my books means it can in theory pick up class levels potentially. A wolf-barbarian would be really interesting to see.
Really, I think the animal checks is worth the price of having to blow an action every turn, and I can't see any reason why a raven familiar can't attack for 1 hp of damage each round, either. But meh, pets may be the way to go. A ring of Speak with Animals might be worth picking up, perhaps to give Advantage on the skill checks, too.
Friendly advice: Few things are as annoying to other people as starting a sentence with "Actually."
I did not say that pets are "prevented" from getting better, only that ranger's companions get better as the ranger levels. Without the DM intervening and making it so, a pet will not. So intervene if you want, nobody is stopping you. You wanted to know, however, if pets were demonstrably better than ranger's companions and short of the DM intervening, they are probably not.
Sorry about that. 
This may take a bit of thought, but as it stands, it feels like having pets may be more to a character's advantage than having an animal companion or familiar. I'll need to see this in play, I guess.
Looking at it from the viewpoint of another player, I would NOT like to see one of the other PCs get a pack of hunting dogs to bring to every fight.
For the same reason that Warlords should not bring a half-dozen men-at-arms, and Necromancers with Raise Dead should not get self-aware undead minions.
I'm going to be getting bored while the boss-plus-followers takes care of his multiple-personality turn.
I've seen players do most of those - characters with enough money to hire henchmen to assist with a dungeon, a summoner bring in a pack of animals by using methods to extend duration, and necromancers who controlled undead to make a pack that followed them around. I consider most of these to be perfectly viable tactics for adventuring.
If the Ranger's companion requires actions to control, I don't see how you could justify allowing other pets not to require an action to control. We'll probably get hireling rules eventually, but in their absence if you have a pack of dogs trained for combat with you (ostensibly uncontrolled), you'd want to count them as additional PCs when determining the XP budget for your fights, and divide the XP gained evenly between the PCs and these NPCs
If I did divide the XP in such a fashion, then I'd also be allowing the animals to gain levels at a normal rate. Not that I see that as a bad thing.
I've tweaked one rule concerning ranger companions - I told the player that it costs him an attack to command the animal to strike at one target, but that continued attacks on the same target don't need his constant watching. That seems to have worked out well.