In order to develop a website where personal height is crucial information, I'd like to know if it is a common feature for every passport to have personal height written in the personal data section.
No, height is not guaranteed to be present.
British passports (I'm sure there are other countries too, but one counterexample seems to be enough for your purposes) do not contain the holder's height. See this sample (found on wikipedia) for an example.
That only shows one page, but I am a British passport holder and can confirm first hand that my height does not appear anywhere in my passport. That information wasn't even requested when I last renewed it (about two years ago).
I have two passports, Australian and USA - neither has my height. My USA driving license did have my height, but my Australian doesn't - come to think of it, I don't think there's any government department in Australia that would have any clue how tall I am, and they definitely don't put it on any ID that they've given me!
Heights can change a lot- even for adults, even during the day. It's one reason why it's often not bothered with as a data point. I saw you're developing some kind of site for athletes, so there may be some consideration for collecting the height of those who sign up, but if you were thinking of having them upload a scan of their passport to parse the information, I'm not sure anyone would want to do that... This is the travel StackExchange site, so certainly in the context of travel, I wouldn't recommend anyone upload scans of their passports to anyone other than for the purposes of obtaining travel visas, or directly related to travel (such as potentially for hotels, airlines, etc.) - even then you should be picky about who you let have it.
I would venture that this is a disappearing information.
I remember that it was common on some European passports (in the 80's, 90's), now it is going away (gone for some).
I believe this is the same information as with eye color - one extra bit of identification when there was no biometrics and the border officer had to do their best to match the person in front of them to the tiny bits of information in the document.