I cannot give a personal account because I have only been to Taoyuan airport (where nobody objected to me taking pictures—in fact, nobody ever objected to me taking pictures at airports even with a DSLR camera).
However, I will note that Kaohsiung airport, the second-largest airport in Taiwan, advertises its observation deck for plane spotting and other activities on its website. It specifically mentions plane spotting as a reason for its existence and its terms of use (same page) do not mention any prohibition on taking pictures (while it does mention, e.g. no vandalism, no smoking, no loud noises, no betel nut chewing).
In addition, I found no mention of photography, pictures or photo (outside photocopy machines); neither by searching the entire site by the search bar supplied, nor in the FAQ section.
This leads me to strongly assume that the claim is not generally true.
That said, it is always a good idea to abide by the following:
- adhere to any signs that prohibit photography. Try to look out for them so that their presence doesn’t surprise you
- do not take pictures of security installations, like the passenger screening facilities
- do not take pictures of security personnel
- do not take pictures of any military installations or equipment or personnel (applies especially if the civilian airport is adjacent to a military air base)
- do not take pictures of customs or immigration
- exercise common sense when judging whether it is permissible to photograph other travellers