So the question in general isn't as cruel as it sounds — requiring the DM to manage all animal companions really isn't necessary (though Handle Animal checks exist and should be used when applicable). The question here stems from when a character purchases an animal to be used as a mount.
For example, a halfling paladin purchases a (combat-trained) riding dog at level 2 to cruise around on as a mount. Should the player know how many hit points the mount has at any given time? Granted, the best players probably could hold onto the sheet and HP and act without the use of metaknowledge of the animals' hit points, but I'm speaking conceptually first.
But if we wanted to be as cruel as the rules can be (despite my introduction) what exactly should allow a player to know the hit points of an animal companion, even if it was granted through a class feature (like a Druid's Nature Bond or a Cavalier's Mount)? A common houserule at tables I've played at is that Heal can be used to roughly gauge enemy hitpoints, but even with that being granted, does it become the only way to do so?