A reason for this would be that many species of birds have pores in the skin below their feathers that release oil. This gives their outer feathers a water-repellent coating that allows them to fly in the rain. However, if the rain were strong enough, it would likely outweigh the oil coating but only (at best) slow it down. This would eventually cause the bird to lose power and have to land, but it still has the ability to fly and has a somewhat decent method of protection against such weather.
Another reason would be that wind has enough force to move rain, as is seen in many storms and also at sea; wind is partially responsible for tidal waves. Although this only applies to ‘wind’ based Pokémon such as Thundurus, Tornadus and Landorus, it is a widely accepted reason since the air and wind is often associated with the Flying type.
On the other hand, birds that have little relationship with water such as garden birds are unable to swim as they don’t have the correct adaptations or requirement to do so. As a result, placing a blackbird in a human bath would likely cause it to thrash about in an attempt to fly away, although doing this could possibly drown it as it would lower its surface area that would be in contact with the surface of the water otherwise.
As for the Bug type, there are a number of possibilities, one of which requires a little biological understanding: some species of insect have specialised cells in their backs that contain more ribosomes than most. Ribosomes are responsible for protein synthesis, which creates proteins that keep the insect warm and therefore evaporate small droplets of water as they land on the bug.
Also, you need to consider how many species of insect (most of which are coincidentally flying creatures) spend the vast majority of their lifetimes underwater. For example, a mayfly spends a whole year as an aquatic nymph before spending just a single day out of the water and breeding before dying.
Other species such as bumblebees have fur-like coats that, similarly to the birds, release oils and prevent water reaching the exoskeleton (skin equivalent) of the insect.
In addition, real-life animals are much smaller than their Pokémon equivalent, and therefore the amount of water that would kill a butterfly would probably feel like drizzle to a Butterfree.
Source 1: a biology lesson last week for the part on birds.
I hope this helped!