As you rightly suggested, the sounds vary depending on the size of the island and how developed it is. An island like Phuket is very big (550 sq. KM) with over 300 000 residents, plus probably twice as many tourists at any one time. Therefore, human-generated noise becomes part of the overall ambiance and also disturbs the natural sonic environment which exists.
One sound that I noticed was very prominent everywhere in Thailand was the cicadas. However, these vary greatly depending on the natural environment they are in. For example, I was on a small, relatively unknown island called Koh Phayam on Thailand's West coast and there was a jungle in the middle of the island. Here, the cicadas were singing in chorus very intensely (there must have been literally thousands of them on each tree). On more built up islands (the touristy Ko Phi Phi Don for example) the cicadas are much less intense. Another factor that makes a difference is the time of day. I noticed that the cicada calls varied depending on the time of day.
In terms of bird calls, I'm no birdwatcher so I can't give you names of species, but I've uploaded a little mix of the various ambiances I captured around the country which features some bird calls as well as the cicada variations I've already discussed. Again with the bird calls, a lot depends on the type and size of the island. I hope it helps.