The Inquisitor Archetype Sacred Huntsmaster gets an animal companion. It can also gain another at level 4 with the Animal domain.
Erastil Diefic Obedience can pick up another one of temporarily.
What I want to know is do they all benefit from the Sacred Huntsmaster abilities like
At 3rd level, a sacred huntsmaster automatically grants her teamwork feats to her animal companion. The companion doesn’t need to meet the prerequisites of these teamwork feats.
or only the one gained from the archetype?
That benefit applies to only 1 animal companion.
There are issues with getting several animal companions; generally, you don't get more than 1 animal companion, however, you can get more than one animal companion if:
1) the animal companion you have is not a valid option for the new animal companion you get [e.g. you have a horse ac, but now you need to choose a bird]
2) if you have the ability to split up your effective druid level via an archetype's class feature.
You can take option 2 with the Cavalier Huntmaster, the Druid Pack Lord, the Hunter Packmaster, the Ranger Beast Master or the Ranger Jungle Lord.
If you want to give your Teamwork feats to more than 1 animal companion, you should take the Hunter Packmaster class, as they allow you to do so. The Cavalier Huntmaster has his Tactician class feature limited to animal companions.
A magical solution is the Shared Training spell (2nd-level spell for hunter, inquisitor, ranger) that allows you to share 1-4 teamwork feats with 1 ally per 3 CL.
Why does Sacred Huntsmaster with the Animal Domain not work?
Why does Sacred Huntsmaster with the Animal Domain not work?
Because both options (1=Sacred Huntsmaster, 2=Animal domain) loop back into druid levels.
1a/ Sacred Huntsmaster levels = Hunter levels:
This ability works as the hunter class feature of the same name, using her inquisitor level as her hunter level.
1b/ Hunter levels = Druid levels:
This ability functions like the druid animal companion ability (which is part of the nature bond class feature). The hunter’s effective druid level is equal to her hunter level. If a character receives an animal companion from more than one source, her effective druid levels stack for the purposes of determining the companion’s statistics and abilities.
1c/ conclusion: Sacred Huntsmaster levels = Druid levels
2a/ Inquisitor levels = Cleric levels:
The inquisitor uses her level as her effective cleric level when determining the power and effect of her domain powers.
2b/ Cleric levels = Druid levels -3:
Your effective druid level for this animal companion is equal to your cleric level – 3. Levels of cleric and inquisitor stack for the purpose of determining domain powers and abilities, but not for bonus spells.
2c/ conclusion: Inquisitor levels = Druid levels -3
You simply gain the same ability levels again, and they mention that these levels stack. So all those levels are added together, and a 10th level (no-multiclass) Sacred Huntsmaster with the Animal domain has an effective druid level of 10 + (10-3) = 17 for his Animal Companion.
However, Paizo has issued a FAQ in regards to this, ruling that you cannot increase your Animal Companion level above your character level +1, so even though that 10th-level Sacred Huntsmaster has an effective druid level of 17, his animal companion is limited to 11th level (=10+1) due to the FAQ.
This arbitrary nerf hits such a build hard, and a good GM would just give you 2 animal companions, since you get 2 effective druid levels every level, but RAW he is not required to do so. :(