Why wouldn't he?
Nothing I can find in the rules says that being possessed cuts off divine powers.
I'm sorry, but the chatterer swarm on Archives of Nethys has no ability to use any spells, spell-like abilities, command the animal companion, or anything else.
Just babble insane commands. The only real info it gives on magic is that it "can also attempt to use the host’s charged magic items."
Are y'all looking at another creature somewhere else?
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As for the animal companion, remember that they are not simply an extension of the PC. Just as a GM can take direct control of an animal companion or veto certain actions for a PC, the same applies to an NPC who might possess the PC. It's why the game has a GM.
For example, the GM should consider whether the animal companion can sense that the PC is possessed, or that something is "wrong". While not reflected in Pathfinder as a mechanical ability, animals are commonly thought of as having unusual senses, like sensing that an earthquake is coming or sensing supernatural events and running away, etc. Even if the GM wants to play it RAW, the animal companion might get a Sense Motive check to sense something is wrong (maybe with a +2 circumstance bonus because it knows the subject so well). In any case, the GM would need to adjudicate how the animal companion reacts - might simply not obey, might become confused, might run away temporarily, might even attack it's master thinking its an enemy.
Even if the animal companion doesn't sense something wrong, the GM should remember that the animal companion isn't an unthinking robot. The animal companion has been traveling with this group of companions for some time, they are all part of its pack, they treat it well, maybe they all give it treats, etc., then suddenly the master orders it to attack it's pack-mates. The GM could rule that the DC on the handle animal check is higher. For some animals, this might be a minimal modifier, but other animals that tend to think in packs (like a dog or wolf) might have a very high modifier. Even if the handle animal check succeeds, the animal might hesitate, losing a round or two, mostly due to confusion, but maybe partly reluctance.
Intelligent animal companions, and cohorts as well, would have to be adjudicated by the GM. The Paladin of Iomedae suddenly attacks one of the party members and orders his junior Paladin/cohort to cut down the cleric of Iomedae for no discernable reason... does the cohort do it? The GM is going to have to do his/her job and adjudicate.