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Personally I'd go with biped they're usually "carrying" the load with their wings whereas a quadraped has all 4 four limbs on the ground to spread out the load. That is dragon standing on the ground has four legs firmly planted and the load spread out across them whereas a dragon flying is supporting all that weight on its wings alone. Even that feels a bit too generous too me as a flying creature shouldn't be able to carry as much as a walking one but raw flying/walking/swimming makes no difference to load so I just count the limbs supporting the load.
Note that "quadruped vs biped" is also a matter of frame, not just how many feet you divide the weight between. Horses actually have some really weak legs, but they have really long bodies so 500 pounds of weight instead handing on two shoulders like it would be a human.
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RAW a hippogriff can carry 198 lbs as a Light load, with no distinction between flying or running on the ground in the text. The creature has a 15 Str and is considered a Large sized quadruped meaning it's carrying capacity should be x3. 15 Str is 66 lbs Light load, multiply that by 3... you get 198. So, by RAW there is no distinction between flying or riding capacity, calculate the carrying capacity as per the type of creature and it's size.
Outside of the Rules Questions sections, feel free to make up whatever carrying capacity for flying creatures you'd like.
Interesting but I'm not entirely sure how much stock we can put in that. Its in the descriptive text yes but its just one example and an example of a flying quadraped at that with no distinction of whether its flying or walking for that load. Other flying creatures including ones described as common companions don't seem to have a carry capacity listed at all Giant Owl's, Manticores, Rocs, etc. What the rules say is . . .
Quadrupeds can carry heavier loads than bipeds can. Multiply the values corresponding to the creature’s Strength score from Table 7–4 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.
So as I said I applied it by counting the limbs carrying the load, 2 limbs = biped, 3+ limbs = quadraped (no rules for 6 or 8 limbed mounts). Most creatures don't describe their carrying capacity (wonder why the hippogriff does?). Otherwise you wind up with a situation where say the Snow Roc (gargantuan, 2 legs, 28 str) has a light load of 400 lbs and a young red dragon (large, 25 str, 4 legs) has a light load of 798 lbs. The weaker and smaller dragon can carry almost twice as much even though they're both flying and not using the limbs in question.
I will allow awesomenessdog's point about frame which I hadn't thought about but still I'd like to see a written carrying capacity for a bird mount to determine if the designers are thinking 4 legs = quadraped = more carrying capacity even when flying or legs + wings = number of limbs = quadaraped for determining it. Otherwise it makes a big difference in the value of mounts as I said above the dragon's weaker but can carry a lot more (not to mention its other abilities) if you can convince one to help.
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"Well, the rules (discounting magic) are an attempt to keep things realistic,"
Not really, no.
"A horse can carry X amount because that's truly about what a horse can carry.... If you throw some wings on a horse you wouldn't expect it to be able to fly with the same weight it can walk with because it's unrealistic."
I nor anyone I have ever played with have ever had this thought. Trying to apply physics to fantasy flying creatures (something something wing span) makes a lot of things fall part.
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DnD3.5 Hippogriff on d20srd Str:18, HD:3d10+9, load:lgt 300, med 600, hvy 900.
DnD3.5 Carrying Capacity
STR 15 -> Lgt:66 lb. or less, Med:67-133 lb., Hvy:134-200 lb.
STR 18 -> Lgt:100 lb. or less, Med:101-200 lb., Hvy:201-300 lb.PF1 Hippogriff on AoN Str:15, HD:3d10+6, load:lgt 198, med 399, hvy 600.
PF1 Carrying Capacity {identical}so you can clearly see the *3 scalar for Large size was used in both cases.
It's rather unrealistic as is the game model.
In game terms a scalar would have to be added to 'natural fliers' and another for 'magical flight', then the spell Fly and Overland Flight examined as spells generally operate off caster level or spellcasting ability bonus. Encumbrance rules for fliers would have to be created which would impact Fly Checks and gaining altitude. If you go to all that trouble then you'll have to check the body mass of natural fliers and lessen it... it's a good topic for the Homebrew forum.
Not saying it isn't there just that so far as I know that's the only flying beast to have that value included Manticores, Snow Rocs, Giant Eagles, Dragons, etc don't list carrying capacity that I can see so its hard to judge if that's a single writers opinion based on "four legs quadraped" or an actual area they've not thought about. See my above example of a smaller, weaker dragon having a bigger carrying capacity than a snow roc if you give the dragon the x3 multiplier for being a quadraped but not the snow roc as its a biped. RAW I could argue the roc's a quadraped as it doesn't have "arms" it has 4 "limbs for movement" wings and legs but well I'm sure you see the issue. Either you apply quadraped regardless of flying mechanic and get weird carry weights or you apply the quadraped rule based on means of travel e.g hippogriff flying = no multiplier, hippogriff walking multiplier.
I don't think they thought about these combined cases with the rule probably RAW vs RAI. Since looking in the player guide, DMG and Beastiary I don't see quadraped defined anywhere and as quoted above it just has a basic quadraped carry more no mention of more or less limbs (snakes) so I think it really is just an area that slipped through wtih no one thinking about it. The made the quadraped rule for land creatures and never considered flying, aquatic or other limb counts.
"Well, the rules (discounting magic) are an attempt to keep things realistic,"
Not really, no.
"A horse can carry X amount because that's truly about what a horse can carry.... If you throw some wings on a horse you wouldn't expect it to be able to fly with the same weight it can walk with because it's unrealistic."
I nor anyone I have ever played with have ever had this thought. Trying to apply physics to fantasy flying creatures (something something wing span) makes a lot of things fall part.
I and I suspect a lot of others have had that thought actually along with other ones like if a piegeon poops on things below what happens when a pegasus does that or do pegasi fly into windows like birds do?
I don't think there is any tether to reality in any of it, honestly. Adult Dragons, for example, would require wingspans hundreds of meters wide to achieve flight without magic... lifting literally nothing more than their own weight. So, I would just go off the creature's size and whether it is bipedal or quadruped. It's just easier that way, and hopefully it ends up being able to carry more... always go for the better option rather than potentially nerfing something for no reason.
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"I and I suspect a lot of others have had that thought actually along with other ones like if a piegeon poops on things below what happens when a pegasus does that or do pegasi fly into windows like birds do?"
I have as well, but these instances are completely different than varying carrying capacity due to body type and movement.