Griffons are Quadrupeds.
Avian refers to the general layout a bird has (two legs, two wings, torso and head), not simply looking like a bird. This can be seen in the examples given:
Axe beak, dimorphodon, dinosaur (pteranodon), dire bat, eagle, giant vulture, hawk, owl, quetzalcoatlus, roc
Bat, dodo, hawk, osprey, owl, parrot, raven, rhamphorhynchus, snail kite, thrush, toucan
A peteranodon looks nothing like a bird, nor do bats, but they have generally the same body plan as a bird (two legs, two wings).
A gryphon has a head, a torso, two wings and four legs, even if only two are clawed like those of a bird (talons) and the other two are clawed like those of a lion.
Looking at the magic item body slots, there is no Wings Body Slot, so any creature with wings can ignore their wings as a valid slot or even ignore the existence of their wings as something that would get in the way of how their magic items work, unless the magic item specifically calls for that (like the Sky Marines Elite Saddle).
The closest thing to a wing slot is the shoulders slot:
Shoulders: This slot consists of capes, cloaks, cords, mantels, pauldrons, shawls, stoles, wings, and other items that can be worn on the shoulders.
Also, based on the monster entry for both the griffon and hippogriff, they both have 3× carrying capacity, which is a benefit from being a quadrupedal creature.
Quadrupeds can carry heavier loads than bipeds can. Multiply the values corresponding to the creature’s Strength score from Table: Carrying Capacity by the appropriate modifier, as follows: Fine ×1/4, Diminutive ×1/2, Tiny ×3/4, Small ×1, Medium ×1-1/2, Large ×3, Huge ×6, Gargantuan ×12, Colossal ×24.
This means griffons should be classified as closely as possible to a Quadruped (claws/paws). This is also the opinion of a PFS Venture Agent.
If semanthics are the problem, remember that talons are a type of claw.
A claw is a curved, pointed appendage, found at the end of a toe or finger in most amniotes (mammals, reptiles, birds).
a sharp, usually curved, nail on the foot of an animal, as on a cat, dog, or bird.