What does the sound "app" mean in Chinese?
I'm pretty sure it doesn't have any meaning in Mandarin itself, given that "words" in Mandarin end with either a vowel, "n", or "ng" (as per the pinyin romanization). It could be another word from English but I doubt it (given that I can't think of any other words with similar spelling/pronunciation in common usage). Moreover, in my experience, most people who speak Mandarin just use "app".

In my experience Cantonese speaking people seem more inclined to use "app" as well, even though there are a couple characters with slightly similar pronuncation (but as obvious, significantly different tones) in Cantonese, such as 鸭 (aap with the jyutping romanization system). That said, given the tonal nature of the words in Cantonese and the fact that there is still a different pronunciation, I don't think there's any risk of confusing "app" with any of those words.

(However, it's worth noting that my experiences have mainly been around Cantonese/Mandarin speakers who probably are inclined to code-switch with English words and phrases than average.)

To conclude, I'd say that while there might indeed be Chinese speakers who pronounce "app" as "A-P-P", I wouldn't attribute this to any words in Chinese with similar pronunciation and different meaning.