I'd say it's because to a Mandarin speaker A.P.P is easier to pronounce than 'app' (Mandarin doesn't have any syllables that end with a p sound, but it does have syllables that start with p).
There's also a history of 3 letter acronyms imported from English, such as DVD, VCD, MP3 which are all pronounced letter by letter and even though APP is not an acronym, to a native speaker it fits a similar pattern.
For your example, people end up pronounce A.P.P because not everyone knows that app comes from `application` and therefore you might run into conversations with people speaking with different pronunciations. Some people pronounce 'app' the normal way, and some people pronounce 'app' similar to (ape).
After awhile, people would explain what they're trying to say by spelling out, and eventually you'd learn to just say A.P.P instead of attempting to incorrectly pronounce the word.
Same thing goes with Facebook, which is referred as F.B, and Instagram, which is usually referred as I.G