DMK (Don Mueang Airport) IS in Bangkok, so yes, the website should say "Bangkok." I will guarantee with 100% certainty that your reservation confirmation and your boarding passes (whether printed or electronic) included the code DMK and not BKK.
There are a lot of things to complain about when it comes to airlines in general, and AirAsia specifically. But not knowing what airport you booked, that's on you.
Their website was correct. Why did you assume Bangkok meant Suvarnabhumi?
If you Google for “Bangkok to Singapore” half of the flights Google will show you are from DMK.
If they’d listed UTP as being Bangkok, then sure, I’d have some sympathy, but referring to DMK as being Bangkok is probably more valid than referring to NEW as New York.
Yeah Bangkok has two airports: Suvarnabhumi (BKK) and Don Muang (DMK). BKK is the airport you land at if you are going to Thailand from Europe or North America, etc. (but not necessarily Asia). DMK is the old international airport that mostly serves domestic flights and a small set of international flights to Singapore, Cambodia, Vietnam. Air Asia only has flights at DMK so when they list Bangkok they are correct.
There are a lot of things to complain about when it comes to airlines in general, and AirAsia specifically. But not knowing what airport you booked, that's on you.