You almost always need a Full-round Action to take multiple attacks
Natural Weapons are not an exception
As a standard action, you can use any of your natural weapons to make a single attack. You cannot make attacks with multiple natural weapons, not even paired weapons like claws. If a weapon is “secondary” (it will say so in its description), it makes the attack at −5 even if it’s your only attack.
As a full-round action, you may attack once with each weapon. Only one weapon or pair of weapons can be “primary;” even if you have more than one weapon or pair of weapons that claim to be “primary” in their description, you must pick one or one pair to be primary, and all others are secondary. Secondary natural weapon attacks are taken at a −5 penalty.
This means your attack routine with two claws (primary pair) and a tail attack (secondary) is +0/+0/−5 at BAB +0.
There are some (rare) ways to attack more than once as a standard action, and some of them are compatible with natural weapons. These will explicitly state this exception to the usual, however.
Natural Weapons and Manufactured Weapons use totally separate rules
Natural weapons never receive iterative attacks, and there are no cumulative −5 penalties, only a single −5 penalty applied to all secondary natural attacks. If you have the Multiattack feat, all secondary natural weapons attack at a −2 penalty instead of a −5 penalty.
None of the rules about manufactured weapon attacks (iteratives, two-weapon fighting, Flurry of Blows, etc.) apply to natural weapons.
Thus at BAB +20, that same pair of primary claws and secondary tail slap attacks at +20/+20/+15, or +20/+20/+18 with Multiattack. If you added a pair of secondary Tentacle attacks, that would be +20/+20/+15/+15/+15 without Multiattack, or +20/+20/+18/+18/+18 with.
Unless otherwise specified by the natural weapon, primary natural weapons add your Strength modifier to damage, while secondary natural weapons add half your Strength modifier to damage.
You can mix manufactured weapons and natural weapons.
When you do so, you gain all your normal manufactured-weapon attacks (including iteratives), plus any natural weapons you have, each as a secondary weapon, with the caveat that you cannot use any natural weapons that are being use to hold a manufactured weapon. I.e. if you hold your sword in one hand and attack with it, you cannot use the Claw attack from that hand. If you instead two-hand a Great Axe, then you cannot use either Claw attack.
When you do this, you can mix natural weapons with full-attacks that use the two-weapon fighting combat option or Flurry of Blows class feature.
Example: Longsword, pair of claws, and tail slap
Thus if you had a Longsword, a pair of Claws, and a Tail Slap, your attack routine would be +0/−5/−5 (sword, claw, tail) for BAB +0. Multiattack improves this to +0/−2/−2. At BAB +20, it’s instead +20/+15/+10/+5/+15/+15 (sword, sword, sword, sword, claw, tail) or +20/+15/+10/+5/+18/+18 if you have Multiattack.
Same character with dual kukris and the Two-Weapon Fighting feat
If you instead have the Two-Weapon Fighting feat, and use that option with a pair of light weapons (call it a pair of Kukris), you attack at −2/−2/−7 (kukri, kukri, tail), since both hands with Claws are being used to attack with Kukris, and the Tail is always secondary when used with manufactured weapons (hence −5; note that Two-Weapon Fighting does not penalize it, however). The Multiattack feat would reduce the penalty, so the tail would be at −2 (+3 attack for one out of three attacks is not a great use of a feat, but just for completion’s sake).
At BAB +15 and with the Improved and Greater Two-Weapon Fighting feats, you get +13/+13/+8/+8/+3/+3/+10 (left kukri, right kukri, left kukri (2nd), right kukri (2nd), left kukri (3rd), right kukri (3rd), tail).
Flurry of Blows modifies your full-attack
It adds additional attacks, and applies a penalty. It is otherwise a normal full-attack like any other.
This means you may use any weapon you have available for any given attack. If you are holding two weapons, you can use one or the other for each attack you’d normally have.
Note that this is exactly the same as when you don’t use Flurry of Blows. If you have multiple attacks for whatever reason (BAB +6 or higher, the haste spell, whatever), you can use any weapon you have available for any given attack as you please.
On the flip side, you do not have any more attacks, normally, than you would with a single weapon. You simply get the attacks you would otherwise have. Those can be the ordinary attacks of your full-attack, or extra attacks from Flurry of Blows, or whatever. Simply having two weapons does not give you extra attacks.
To get extra attacks, you have to use the Two-Weapon Fighting combat option (and then you are subject to its penalties and restrictions). In particular, Two-Weapon Fighting is an exception to this “any weapon for any attack” bit. Note that this is more than just having two weapons; you have to actually choose to use Two-Weapon Fighting, or none of the following applies. When you do choose to use Two-Weapon Fighting, it changes your full-attack (much as Flurry of Blows does): it grants an extra attack and applies a penalty, and it also applies a special restriction on which weapon you can use for which attack: specifically, your bonus attack from Two-Weapon Fighting has to be a different weapon than the one you are otherwise using.
Furthermore, under Pathfinder rules, Flurry of Blows is Two-Weapon Fighting (except it doesn’t actually require you to use two weapons). As such, the idea is you cannot use it and Two-Weapon Fighting at the same time. If this is the case, there is absolutely zero benefit to having two weapons, except I guess the ability to pick which one you want. You’re better off with a more-powerful single weapon than splitting your wealth between two.
Unfortunately, the rules here are notoriously vague and have required numerous amendments, explanations, Paizo contradicting themselves several times, and finally overruling previous rules entirely because they never made sense. This is actually one of the most famous fiascoes in Pathfinder history, and the rules are left a mess as a result. I suggest just returning to the 3.5 rules, before all this started: Flurry of Blows and Two-Weapon Fighting are completely separate things, combine them if you want (taking even more penalties and gaining even more attacks). This is still a bad idea, since it is a very weak option, but at least the rules are clear, make sense, and there’s actually, theoretically, a benefit to dual-wielding as a monk.
Best Answer
Yes, this is (one of) the main purpose(s) of the Feral Combat Training Feat.
By selecting Claw for Feral Combat Training, you gain the ability to use it as your weapon (similar to wielding a 'monk weapon') for the subsequent extra attack. The extra attack could be your Claw or an unarmed strike per the normal monk rules.
This differs from the other question because the other question wanted to use a full set of extra attacks; in your case, you are merely substituting your Claws as a valid option for the Flurry of Blows strike.
To clarify, this does not allow you to Full Attack with Natural Weapons+1, but instead allows you to use the benefits of your Monk class with your Claw(s) and the benefits of your Claw(s/Blades) with your Monk class.
Per this FAQ:
To summarize: