SRD said:INA: Choose one of the creature’s natural attack forms. The damage for this natural weapon . . .
The reason it says this is because the majority of creatures have claws and a bite, and you have to choose if you are better with claws, or with biting. Or in the case of dragons, wing buffets, tail slaps, slams, and what have you. Improved Natural Attack, like Weapon Focus can be selected more than once, but it does not stack. Each time you must select a different Natural Attack to improve.
As a Dire Tiger, you take Improved Natural Attack: Claws to gain the benefit with all your claws. If you want to improve your Bite, thats INA: Bite.
This is functionally identical to taking Weapon Focus: Longsword and Weapon Focus: Flail.
The only possible point I can see from your angle is that "Claws of the Beast" arent the same specific claws that you took the feat with, but I see nowhere in the feat that says you can only use the feat with a certain specific claw, simply that your claw attacks go up one step in damage die
SRD said:Improved Natural Attack [General]
Prerequisite
Natural weapon, base attack bonus +4.
Benefit
Choose one of the creature’s natural attack forms. The damage for this natural weapon increases by one step, as if the creature’s size had increased by one category: 1d2, 1d3, 1d4, 1d6, 1d8, 2d6, 3d6, 4d6, 6d6, 8d6, 12d6.
A weapon or attack that deals 1d10 points of damage increases as follows: 1d10, 2d8, 3d8, 4d8, 6d8, 8d8, 12d8.
This feat may be taken multiple times, but each time it applies to a different natural attack.
You cannot, by the text, even take the feat again for "Claws of the Beast" so I have got to assume the feat applies to all claw attacks you are capable of making.