The Pathfinder 2e rules are quite clear about how a character can issue orders to a normal animal, and what that animal can do.
The Command an Animal (Core Rulebook pg. 249) Nature skill action states,
You issue an order to an animal. Attempt a Nature check against the animal's Will DC. [...]
Most animals know the Drop Prone, Leap, Seek, Stand, Stride, and Strike basic actions. If an animal knows an activity, such as a horse's Gallop, you can Command the Animal to perform the activity, but you must spend as many actions on Command an Animal as the activity's number of actions. You can also spend multiple actions to Command the Animal to perform that number of basic actions on its next turn; for instance, you could spend 3 actions to Command an Animal to Stride three times or to Stride twice and then Strike.
- Success: The animal does as you command on its next turn.
To get an animal to perform actions other than those bolded above, you must train it using the Train Animal (Core Rulebook pg. 268) downtime activity granted by the Nature feat of the same name.
You spend time teaching an animal to do a certain action. You can either select a basic action the animal already knows how to do (typically those listed in the Command an Animal action) or attempt to teach the animal a new basic action. [...] It’s usually impossible to teach an animal a trick that uses critical thinking.
- Success: The animal learns the action. If it was an action the animal already knew, you can Command the Animal to take that action without attempting a Nature check. If it was a new basic action, add that action to the actions the animal can take when Commanded, but you must still roll.
So far this is clear. But Animal Companions (Core Rulebook pg. 214), as granted by certain classes and archetypes, work differently.
Your animal companion has the minion trait, and it gains 2 actions during your turn if you use the Command an Animal action to command it; this is in place of the usual effects of Command an Animal, and you don’t need to attempt a Nature check. [...]
Your animal companion uses your level to determine its proficiency bonuses. It’s trained in its unarmed attacks, unarmored defense, barding, all saving throws, Perception, Acrobatics, and Athletics. Animal companions can’t use abilities that require greater Intelligence, such as Coerce or Decipher Writing, even if trained in the appropriate skill, unless they have a specialization that allows it. [...]
The Support Benefit entry indicates a special benefit you gain by Commanding the Animal to use the Support action. The Advanced Maneuver entry indicates a powerful new action your companion learns how to use if it becomes a nimble or savage animal companion.
Some specific types of companion have additional proficiencies, either Intimidation, Stealth, or Survival, and with additional feats those skills can increase as far as Master rank.
But are the actions of an Animal Companion strictly limited to the following list?
- The basic actions listed in Command an Animal;
- The Support and Advanced Maneuver actions for its type;
- Additional basic actions that its owner adds via Train Animal.
I can't see any rule that says otherwise, but it leads to some strange consequences.
First of all, the list of actions in Command an Animal has some strange omissions. An animal can't be commanded to Escape from a grapple, and since Fly, Climb and Swim are separate actions from Stride, RAW a bird will run to its destination rather than fly there.
Secondly, the additional skills of Animal Companions are of very little use unless the animal can perform related actions, such as Demoralize, Hide or Sneak. Some skills might have passive benefits, such as if an enemy forces an opposed check, but I think that is rare.
Third, the rules say "Animal companions can’t use abilities that require greater Intelligence ..., even if trained in the appropriate skill", rather than simply reminding us that they can't use Skill Actions.
Fourth, some Animal Companions lose abilities when they become companions. Compare the standard Riding Horse (Bestiary pg. 209), whose Gallop ability was explicitly called out as an option in Command an Animal. But the Animal Companion version of a horse features Gallop as its Advanced Maneuver, which the player must take feats to unlock. The same applies to a gorilla's Frightening Display (normal (Bestiary pg. 23), companion (Advanced Player's Guide pg. 144)). It's worth noting that Frightening Display reads:
Your ape makes a loud and frightening display, throwing a foe off balance. The ape attempts to Demoralize the target creature; this display gains the visual trait and doesn't require language. While frightened by this ability, the target is flat-footed to your ape.
This explicitly gives the ape the Demoralize action; but there are enough extra effects that I am unsure whether that should support or contradict the idea that other Animal Companions can Demoralize.
Fifth (similarly to the Ape example), the Ambusher specialization allows an Animal Companion to "use a Sneak action even if it's currently observed," implying that other companions could Sneak if they were unobserved (as usual with stealth).
Maybe it's the intention that GMs decide what actions are reasonable for particular animals, but perhaps there is some rule that I am missing.