Rattan is strong, and doesn't splinter. A rattan shield, whether bound with rattan skin or silk thread, will resist splitting along the grain due to the binding. Is there any reason to think that a rattan shield won't be as strong as a wooden shield of the same thickness (or the same weight)?

If the typical rattan shield was very thin and light, then it might be surprising to find that they are very strong. But the typical rattan shield weighs about 3kg - the British Museum Asian rattan shields average 83cm in diameter, and 3kg in weight. The main rattan coil is made of rattan about 9mm in diameter (8-10mm is most common). At this thickness and weight, a lot of strength should be expected.

Were they actually "nearly musket proof"

It isn't clear what that source means by "nearly musket proof". Stopping everything short of a musket? The shield + something else stops musket ball? Supposedly rattan shields backed by cotton padding (how much?) could stop musket balls. Either way, the shields, without augmentation like this, were not musket proof.

I would just think they did the job, but couldn't possibly be more durable than thick wood or iron with hides etc. right?

It should be less susceptible to splitting than wood, so could be more durable than wood of the same weight.

As for rawhide, shields of similar thickness to the typical rattan shield (approx 1cm thick) have a similar reputation for being "nearly musket proof". Both North American and Indian rawhide shields of such thickness have been described in such terms.

Since "nearly musket proof" means "not musket proof", and other (non-metal) shields of similar weight and thickness were similarly very protective (but short of musket proof), I don't think there is any mystery needing explanation.

The term 'almost musket proof' is very vague and it also conveys that they weren't able to stop musket projectiles. So, no, they weren't bulletproof.

Vague writing aside, we can be almost certain that rattan shields and many types of armour used in China in the Ming period were not able to stop musket bullets thanks to the writings of contemporary commander Qi Ji Guang (Qi Jiguang) who explicitly stated in his 'jì xiào xīn shū' (lit. 'New Book on Discipline Effect', also known as 'New Treatise on Military Effectiveness') that common cloth and leather armour as well as wooden and rattan shields are ineffective against Japanese firearms. Qi Jiguang himself was also a big supporter of adopting firearms into military tactics.

Thick layer of lacquer would not add substantial protection as dried natural lacquer is brittle. Rattan, on the other hand, is very elastic and durable while the construction facilitates dissipation of the applied force, making the shield more than adequate to deflect blows, especially when the middle section was reinforced with metal boss. But the impact force of the bullet shot from early-16th century musket or arquebus would be most definitely too much. Of course, a distance is a major factor here and it possible that such shield could stop a bullet fired from far away.

I would also like to point out that Chinese experimented with bulletproof shields and none other than aforementioned Qi Jiguang proposed two designs: one was ruǎn bì (soft wall) made of several layers of cotton cloth over wicker and planks and used as pavise for shooters, the other was gāng róu pái (rigid-elastic shield) made of multiple layers of silk and paper sandwiched between thick layers of cowhide on a bamboo frame. Author attests that the latter was able to stop a musket shot up to 40-50 paces. So yes, Chinese had bullet-proof shields in that period, but these. were not rattan shields.

It is worth noting that almost identical shields were pretty popular in Eastern and Central Europe until early 18th century, especially in Ottoman Empire, Poland, Hungary and Persia (usually knows under its Turkish name 'kalkan' or a name derived from it). Contemporary sources and modern experiments show that despite low weight (3 pounds for shields with metal boss) they were very effective at stopping arrows shot from recurved Mongol bow. That said, Polish accounts from the 16th and 17th century state that Tartars using similar shields and wearing light armour were basically defenceless against firearms.