By the rules, the only Animal Companions that can exist, are those that have Animal Friendship spell cast on them, or those under the control of a class that has Animal Companion class ability like Druid or Ranger. Animals trained by fighters, even if well trained (high animal handling skill ranks) are never Animal Companions.
But that is a non-answer. The question is not whether a pet is an 'Animal Companion' under the strict rules definition of that.
The question is, "Suppose I raise a puppy and train it to be a loyal companion and attack dog. How should one rule?"
To say that the pet is not an Animal Companion doesn't tell us anything. Are you suggesting that pets can't exist? Are you suggesting that by the rules, unless one has an Animal Companion and the appropriate spell that one can't have a pet? Most GMs run a game of D&D with a certain amount of verisimilitude to reality. The expectation is that, unless it is explicitly called out otherwise, that a player can expect that most actions that they would attempt work more or less like they would in real life. This is a necessary assumption to allow players to propose mundane interaction with a setting - objects can be lifted, turned over, doors can be opened and closed, food eaten, ect. This assumption allows all sorts of proposition resolution that would otherwise be impossible to be described in the rules themselves. The rules don't need to describe something as basic as 'objects are tangible and can be grasped', because this isn't a computer game.
And in real life, animal companions (small letters) are known without the use of an Animal Friendship spell. In a game world, the proposition by a player that they acquire a dog and raise it is reasonable for setting based on the expectation that pets are tamable without the use of magic. Saying that the animal companion is not an Animal Companion doesn't answer the question, "What is the status of this animal?" If it was obviously an Animal Companion, there would be no need to ask about the status.
In general, I'd say a 'pet' is like an Animal Companion in every way that isn't magical, but is generally restricted to being a normal domestic animal* and is not so easily replaced or trained. In general, my expectation is that a campaign without a lot of downtime, a lost pet is not replaceable. Creating a 'pet' relationship requires the expenditure of months or even years of downtime depending on the species. Training a pet while adventuring would be next to impossible, and at the very least would be much slower. In general, I find that often players start with the desire of having a pet, then eventually the pet gets killed and is never replaced. Later on, the players may purchase trained animals (usually steeds) but these 'pets' will not have the same loyalty that the starting pet had, and so - for example - are generally unreliable except when directly supervised. Left alone for a significant amount of time in unfamiliar surroundings they'll wander off, lose focus, become frightened or confused, and generally get into trouble. Only an extended period of familiarity can make an animal reliably loyal the way a spell can do in an instant.
In games without bounded accuracy, this has worked pretty well, as eventually things like men-at-arms and guard dogs become as much liability than assistance. One problem I foresaw when we first started getting 5e previews of monsters is that even mighty monsters are much more vulnerable to masses of 'commoners' than they were in earlier editions. So, I'm not sure how well this approach will still work, and if it doesn't what should be done about it. For 3.X though (and 1e) it works just fine.
*(I'd allow unusual creatures as pets only if raised from birth/hatching, only if of similar intelligence to an animal, and only if the trainer succeeds with more difficult skill checks.)
