So, I just made a Simic Hybrid Fighter-Monk, choosing grasping appendages as the 5th-level race augmentation. Upon considering the possible applications, the question occurred to me: Can I perform Flurry of blows with the claw appendages given by the Simic Hybrid Race Trait. And if so, how does it affect the mechanic?

Unarmed strikes can be done with any part of the body. They can be headbutts, shoulder charges, fist punches, kicks, belly slams, anything. As such, having extra arms/claws/hands/tentacles/pseudopods/legs/whatever doesn't change the FoB mechanic. You already have more appendages to attack with than you need anyway.

i was referring more to whether having claws/tentacles applies any kind of 'weapon' attribute to the attacks

Literally what it says. 

• Grappling Appendages - You have two special appendages growing alongside your arms. Choose whether they’re both claws or tentacles. As an action, you can use one of them to try to grapple a creature. Each one is also a natural weapon, which you can use to make an unarmed strike. If you hit with it, the target takes bludgeoning damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike. Immediately after hitting, you can try to grapple the target as a bonus action. These appendages can’t precisely manipulate anything and can’t wield weapons, magic items, or other specialized equipment.

FoB lets you make two unarmed strikes for a Ki point. The appendages can be used to make unarmed strikes, so can be used for FoB. If that's what you do, you can use the special damage and you have the bonus action grapple attempt after hitting. That's it. They're unarmed strikes, so Martial Arts applies.