First of all, unarmed strike can't be any body part you want. It's undefined for non-monks, but the monk class specifically calls out "a monk's [unarmed] attacks may be with fist, elbows, knees, and feet," so it's reasonable that non-monks have to follow a similar restriction (otherwise the non-monk has more versatile unarmed strike options, which is silly).
So a human holding a barrel with his could make a kick or knee unarmed strike (his arms are busy holding the barrel, so no fists or elbows). If he had TWF, he could make two unarmed strikes, one with the left leg and one with the right leg. We know he has to use separate limbs because the revised monk flurry ruling says a monk can use one weapon for all of a flurry, which means it that a special rule for monks and non-flurrying non-monks can't do that.
Now replace the TWF human with a TWF tengu, and get rid of the barrel. The tengu can make two unarmed strikes per round (one with the left leg and one with the right leg), and gets to add his three natural attacks as secondary attacks. That's 5 attacks per round, total. Overall, that puts his attack bonuses at –2 kick/–2 kick/–5 bite/–5 claw/–5 claw. This is a legitimate attack routine.
Now give him the vestigial arm discovery x2. Instead of making two kick attacks, he's making two manufactured weapon attacks. So his total attack routine is weapon/weapon/bite/claw/claw. That's 5 attacks per round, total. This is a legitimate attack routine.
Tengu-A, the two-armed tengu, is making 5 attacks per round.
Tengu-B, the four-armed tengu, is making 5 attacks per round.
Both are valid. The second tengu is not getting any more attacks per round than the first tengu. In other words, the second tengu is not getting any extra attacks per round compared to the first tengu.
Or, to quote the FAQ:
It means "extra," as in "more than you would be able to make if you didn't have that discovery."
Haste and other special effects, if applied to both example characters, should affect them equally. Frex, haste on Tengu A gives it an extra unarmed strike attack per round, haste on Tengu B gives it an extra manufactured weapon attack per round, but both characters are still getting the same number of attacks per round: 5 + 1.
Q: How many attacks per round can the tengu make without the discovery?
Q: How many attacks per round can the tengu make with the discovery?
Q: Is the tengu with the discovery making any extra attacks per round compared to the tengu without?
Q: So this doesn't violate the rules?
Q: What if the four-armed tengu also wants to make unarmed strikes in addition to the two manufactured weapon attacks and the three natural attacks?
A: It can't, because that would put it at 7 attacks per round, which is more than the (two-armed) tengu's 5 attacks per round.