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?
Best Answer
Yes, a player should know their mount's exact HP.
It's so clear-cut that even if the rules clearly said otherwise, you would want to override them to work differently.
I have never yet seen a DM with enough bandwidth to manage all the things which they already need to keep track of (myself included) and every addition to that reduces the quality of DMing they provide. Some cope with it well, but even so, the more you can reduce that burden, then better the game will be.
At the same time, there is a very short list of situations where keeping players in the dark about mechanical features will improve your game, even if the GM has infinite bandwidth. It inevitably feels arbitrary and capricious, and the list of games where that's a good thing is approximately [Cthulhu, Paranoia, End-of-list].
Similarly, you should write down initiatives someplace visible, and allow players to zero in on ACs after a few attempts. Your players are already tracking these things anyway. If you pretend game states are mysterious, they'll play along to humor you. But after a few attacks, everyone knows darn well that the beastie's AC is either 21 or 22, and the Mother-May-I ritual of "Does a 26 hit?" loses appeal fast.
Source: have played in multiple games on each side of the table in each situation. This isn't even close.