so far from what I gathered pets are basically toady giving you companions from beings that can't talk so you end up with the same hardships one would get from trying to order around an reanimated corpse.
that said you could use them as gifts and technically... there is a pet option that can talk ala the gremlins you can acquire and buy loads of them in bulk if you want to have a party of folks that can hold a conversation and probably hold an aim a crossbow or weapon.
but yeah mount wise they aren't 'give me a boost in speed' option and more you get to ride around the back of a being that you have to struggle to keep them from doing what ever they want while following you.
ok had to check it seems like pets start with no civ ids or are part of any civ, they gain a historical figure and the only hf link they get is that the adventurer is their owner. there's no 'you're my pet' hf link for the adventurer so that explains the not knowing of the pet. don't know if talking to them will fix that but this explains that comment about not knowing... since yeah they literally don't know anything about the pet on start up of the adventurer.
like the funny solution to these mount woes was I think solved via modding by ... making the pets talk. which I remember was how I got pets to work before toady added them in by having an interaction that lets you communicate with any creature that doesn't have can speak then giving them that option,
I think the reason ambusher doesn't level on mounted because you're controlling the mount thus you control the movement options and one of them is go in ambush which doesn't really send you the rider in ambush just the mount so what you're doing is training your mount those skills while your adventurer is just riding on top like a baby being carried around by a sneaking mother.
so to train ambusher yourself you need to go into ambush on foot, then hop on the mount then go ambush mode on the mount.
so far on my companion studies followers that you some how ditch will try to follow you to rejoin back to the group and the few moments where this doesn't happen is if you lose someone on a site as doing so causes them to settle into that site which I guess with pets means they either settle in and become part of the livestock or something else.
though I usually go for companions that can handle themselves so I don't have to worry about them horribly dying... which often means getting the large animals.
Have messed around with the order function a bit and it still leaves a lot to be desired. Might be a limitation of the game though, since I noticed my adventurer's horse wanders around the area I have claimed and built a small cabin as a establishing base of operations. The horse will wander around the site, even after being put into a paddock and being ordered to stay and then letting the building function do it's thing. Thankfully it does stay around the site, but still would be nice to be able to hitch mounts and pack animals to something while murderspelunking.
ok so here's the big oopsy I think you need to zone the companions to the site to get them to stay in that spot and I think the game doesn't really let you do this with non talking sapient(?) creatures so you're pretty much telling your pet to wait on your camp and the game just settles them in or pretty much makes them an 'army' group to wait for your return.
so it feels like there something in place that was stopping the ability to get non sapient beings to be zoned in camps either some check to stop the list of companions to get cluttered from all the zombies adventurers might have way back, or some bug fix that meant removing this from the player's grasp or what not.