Get uncommon natural weapons

An Unarmed Strike is (kind of)¹ a natural weapon.

Not all natural weapons are Unarmed Strikes.

Natural weapon is a category of weapon, which includes Bites, Claws, Slams, Stings, Tail Slaps, and yes, Unarmed Strikes (sort of).¹ Improved Unarmed Strike improves only Unarmed Strikes, not the rest. Improved Natural Attack can improve any one natural weapon, except not Unarmed Strikes because Paizo hates Monks even more than Wizards did.

Note that Feral Combat Training changes this somewhat, however. Improved Unarmed Strike itself still doesn’t apply to natural weapon other than Unarmed Strikes, but anything that has Improved Unarmed Strike as a prerequisite does apply to the chosen natural weapons when you have Feral Combat Training, even if usually they only apply to Unarmed Strikes.

Ultimately, you’re burning two feats to absolutely no effect other than possibly making other feats work. If you found some really good feats that improve Unarmed Strikes, it might be worth it, but I doubt it. That’s a ton of feats. And honestly, I think a good DM should just let most Unarmed Strike-improving feats work for other natural weapons; this is not exactly a high-power strategy here. A two-feat tax is wholly unnecessary.

¹ I say “kind of” because Unarmed Strikes are really weird. They mostly follow the rules for manufactured weapons, not natural weapons (e.g. you get iteratives, you don’t get multiple Unarmed Strikes, etc.), but most spells and feats treat Unarmed Strikes as natural weapons. And on top of that, Paizo put a further restriction on Improved Natural Attack to prevent it from working with Unarmed Strikes. So Unarmed Strikes are very much a weird hybrid case. In any event, Claw attacks are not Unarmed Strike attacks.

In order to use all three, the cat must take a Full-Attack action, which takes a Full-Round Action (Move+Standard), which usually means it may not move aside from a 5-ft. Step. That's generally a pretty big limitation on most things that rely on lots of attacks.

Of course, most cats have Pounce, which means they also get a Full-Attack at the end of a charge, but that's a special case. If the druid's cat has that, it can either 5-ft. Step and Full-Attack (nearby target) or it can Charge and Full-Attack (target at least 10 ft. away). Pounce is fairly potent.

Without Pounce, or without Charging even if you have Pounce, if you use a Move action to move, you may only use a Standard action to attack. The Attack action, taken as a Standard, is only a single attack so the cat would have to choose one claw, or the bite. Without Pounce, Charges are also just a single attack, albeit at a +2 bonus.

In any event, yes, all three are Primary attacks and get full Strength to damage.

And of course it's ridiculously powerful; that's the druid for you. The druid is one of the most powerful classes in the game, particularly at low levels, for exactly the reason you describe: the druid gets a pet that can give a fighter a run for his money, and then he gets his own turn.

Unfortunately, it's very, very difficult to address the balance problem inherent in 3.PF. You could nerf the Animal Companion and the druid would still be one of the most powerful characters because of his spellcasting. One step I do like to take, though, is to swap the Animal Companions of the druid and ranger. The druid is the far more powerful class, so the ranger could use it, plus as someone who is in theory the better melee warrior (though that's not actually true thanks to Wild Shape), it makes sense that the ranger would have the bigger Animal Companion to fight alongside.