I'm not seeing any stats in the PHB. What is the limit that saddle bags on a horse can carry? If there's not any definitive stats would it be OK to assume that each saddlebag is similar to a backpack?
I couldn't find any definitive stats either, but given the cost and weight, you should assume roughly double the capacity.
For the math on why it's double despite not being double the weight, essentially it's because volume increases faster than surface area. If you assume the same thickness and density of material, you can use the weight of the object as an estimate of it's surface area. You can then solve for the sack's volume by assuming it to be a sphere, and while it's impossible to know the units, the volume you get with a surface area of 8 is 2.14 compared to 1.05 from a surface area of 5.
Barring further rulings on the matter (such as from the DMG), I'd say it's quite reasonable to project the carrying capacity of a backpack onto a saddle bag. You could definitely account for them being bigger as well, but realistically, a horse's carrying capacity is between 2/3 and triple that of a typical adventurer (480 vs 150-300). Thus if you take the weight of an adventurer, and then add on his gear, and then add saddle bags, you aren't going to want much more than the volume of a backpack additionally attached.
That said, an unladen horse may be able to carry more, but the saddle bag still has physical limits. I'd set these either at or slightly above that of a back pack. It's definitely in the realm of DM discretion though because the rules are silent. Whether he chooses a verisimilitudinous answer or an abstraction will of course depend.
Having Ridden horses and dealing with this... @Glenatron was right. They are pretty small. A saddle bag is usually made of tough material, heavy/thick leather and would account for being heavier than a normal backpack that you might wear. However, they don't carry much in regards to space. So about the size of a satchel for bulk. About 1 Cubic foot total for both sides (if my memory serves.) They were about 1'x 1'x 3-4"... so combined it would be about 1' x 1' x .5'. So you can take those dimensions... or make them larger. You can go from a simple small (as described here) pouch that can probably hold about 15 lbs, or go larger with the saddle backs that have the Saddlebacks on the side AND a Cantlebag on top between the two that can be the size of a backpack for a bit more. You can look this stuff up, as I did to confirm my memory, at Tack Suppliers and Outfitters. It is of course all up to the DM and players to discuss this. But with my game, we have had several riders or people that grew up around farm/ranch areas in their younger years and see them as small satchels and not FULL carrying items. Thus they will eventually look for carts and other things until they cross paths with a "bag of holding" options.
So based on a real saddlebag, measurements of 12" x 8" x 2.25" (meaning 216 cubic inches, or 0.125 cubic foot space) per bag (there are 2 bags per "saddlebags" one for each side of the horse/animal, otherwise it would say "saddlebag" in the PHB) Meaning roughly together they can hold 1/4 a cubic foot of material, or 7.5lbs.
Now, this could be increased for a 'larger carry' bag style (because there are many styles of saddle bags) I still can't see them being more than 1/2 a cubic foot each, and you get 2 when you buy "saddlebags" from the PHB for 5gold, 8lbs of weight, including straps and probably brass fittings.
So "saddlebags" as a single item - I would say 1 cubic foot/ 30lbs of gear. Each individual bag carrying 1/2 a cubic foot, 15lbs of gear.
The weight would be to the extra straps and other materials used to both hold the items together through the travel, as well as strap it to the horse/pack animal to keep them in place.
This also helps as "Bag of Holding" and "Bag of Devouring" are very similar style to 'saddlebags' - they are basically simple large purses. Useful if you plan to make a bag of holding/devouring with Artificer or other means. (as a sack and backpack are no where similar to them)
SO IMHO, as a DM I have stated a saddlebag carries 1/2 a cubic foot, 15lbs and you get 2 when you buy "saddlebags" as well as strapping, brass fittings and the like to enable the ability to secure it on an animal (which, if customized to purses means you can make 2 nice belts, straps for 2 purses/mail bags, and the purses/mailbags themselves) Which, 1 purse/mailbag can be made into a bag of holding (or devouring) as the primary material to do so - either by Artificer means, or by Magical Item creation.
So to the crafty adventurer, saddlebags are 1/2 the size/carry capacity of a backpack, but, also come with a sturdy belt, all matching in brass fittings/fasteners.