Q:

How do animal companion restrictions/bonuses work when multiclassing?

From the druid's Nature Bond ability:

... If a character receives an animal companion from more than one source, her effective druid levels stack for the purposes of determining the statistics and abilities of the companion. ...

But what if one of my sources has restrictions on animal types or gives additional bonuses to the companion?
Suppose I have a level 10 druid with a giant centipede companion. In a fit of madness, I decide to multiclass into oracle and take the Bonded Mount revelation from the Nature mystery:

You gain the service of an unusually intelligent, strong, and loyal mount. The creature must be one that you are capable of riding and is suitable as a mount. A Medium oracle can select a camel or a horse. A Small oracle can select a pony or wolf, but can also select a boar or a dog if she is at least 4th level. This mount functions as a druid’s animal companion, using your oracle level as your effective druid level. Bonded mounts have an Intelligence score of at least 6.

Does my giant centipede continue advancing as I gain levels in oracle, even though it's not on the list of animal types an oracle could select?
Does my giant centipede's intelligence immediately increase to 6 when I gain the revelation?
If the giant centipede isn't compatible with the oracle animal companion feature, what if I chose a horse as my druid companion? Would its intelligence immediately increase to 6 when I gained the revelation?

A:

If you pick a companion that simultaneously satisfies all the restrictions of both class features, then they stack.
Yes, in this case your companion's intelligence suddenly increases to 6. How? dunno. But the Oracle class feature says it has Int 6, so it does.
If you pick a Druid companion that isn't also a valid Oracle companion, then they don't stack, and don't interact in any way. Instead, you have two separate companions: one giant centipede that only scales with your actual Druid levels and only has the benefits of being a Druid companion, and one horse or whatever that only scales with your Oracle levels and only has the benefits of being an Oracle mount. The horse is smart and the centipede isn't.
Ruling taken from this FAQ (which is about Cavalier, but Cavalier uses the same phrasing as Nature Oracle, so clearly the FAQ applies to Oracle too):
Cavalier: Do animal companion levels from the druid class stack with cavalier mount levels?
If the animal is on the cavalier mount list and on the list of animal companions for your other class, your cavalier and druid levels stack to determine the animal's abilities. If the animal is not on the cavalier mount list, the druid levels do not stack and you must have different animals (one an animal companion, one a cavalier mount).
For example, if you are Medium druid and you choose a horse companion, levels in cavalier stack to determine the horse's abilities. If you are a Medium druid and you choose a bird companion, levels in cavalier do not stack to determine the bird's abilities, and you must choose a second creature to be your mount (or abandon the bird and select an animal companion you can use as a mount).
This same answer applies to multiclassed cavalier/rangers.
(Note that the design team discourages players from having more than one companion creature at a time, as those creatures tend to be much weaker than a single creature affected by these stacking rules, and add to the bookkeeping for playing that character.)

