The spell doesn’t say. There might be several places to get some guidance.
- The carrying capacity rules on PHB p176
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.
- The levitate spell says 500 lbs
There’s no clue as to why levitate includes a weight limit and fly doesn’t.
- Carpets of flying (DMG p157) have various weight limits
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.
- Quaal’s Feather Token (DMG p188) can carry 500 lbs
So, in the end, the gm will need to make a ruling based on the specific situation.