Since it’s a df ability I guess it’s about how strong your haki is and I’ll be bold and say that doffys haki was more superior
Superior enough to make some strings, that he wasn't even controlling directly like the ones he used to fight, durable enough to resist to attacks from two of the strongest swordsmen in the series?
I find that very strange.
It being a result of a comparison of their haki would make sense if it were a direct clash between an attack from Doflamingo and an attack from Zoro, but that wasn't the case.
There was no armament haki covering the bird cage to strengthen it.
And what I really find strange is that Oda didn't write Zoro lamenting his lack of power for not being able to cut the bird cage. That was very badly written considering the character's personality. Instead of making Zoro remark how it was rather strange if the bird cage couldn't be stopped (when everyone else was taking for granted that the bird cage couldn't be stopped), the much more natural reaction for Zoro would be say that there's no way he wouldn't be able to cut some strings, and then if he really ended up not being able cut it, his natural reaction would be to lament his lack of power.
Oda preferred to avoid this drama, for whatever reason, so he conveniently made the characters only talk about whether or not it could be stopped, without anyone raising the question about whether or not it could be cut or destroyed.