A caster casts fly on himself. How much weight can he carry? The spell doesn't say.
I am hoping for some rules-based guidance, but what has worked for you would be useful as well.
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There are no special rules for a flying creature's carrying capacity, whether flying naturally or via a spell, so you just follow the normal rules found on page 176 of the PHB.
Q: What are the rules for creatures carrying friendly PCs? e.g mage polymorphed into giant eagle carrying allies? (5e)
A: See "Lifting and Carrying" in the Player's Handbook (p. 176) for rules on carrying capacity.
Q: Are we to assume no adjustment for flying vs walking?
A: The carrying capacity rules make no distinction between walking and flying creatures.
Even though there's not mention of magical flight, there's still no other special rules to govern that.
The spell doesn't say. There might be several places to get some guidance.
On the one hand, this makes a certain amount of sense, especially lacking any other rule. On the other hand, carrying capacity refers to physically carrying stuff around, using muscle power. The fly spell, on the other hand, is magically powered flight. There's nothing to assume it relies on the flyer's physical attributes at all.
There's no clue as to why levitate includes a weight limit and fly doesn't.
The various carpets range in speed and capacity, with capacity ranging from 200 to 800 lbs. They can also carry over their capacity at a reduced speed.
So, in the end, the gm will need to make a ruling based on the specific situation.
Fly simply gives you a flying movement speed, the same way an owl or an Aarakocra have. Beyond that, your ability to carry stuff is the same as it was for walking. If you're carrying more than your strength score allows then you're encumbered, as per the standard rules in the PHB.