I don’t understand this. What practical difference does it make making the time to round to the nearest quarter of an hour instead of the nearest hour? Personally, I don’t care if noon (sun is in zenith) happens half an hour before 12:00 or half an hour after.
Why do such time zones exist?
Well, I do not know the answer to that, my guess is that it is for historical, political, geographical, and socio-economic reasons.
For example in terms of India, they had two timezones before they adopted a compromise: UTC+5:30.
Nepal uses UTC+5:45, partly to distinguish itself from Indian Standard Time, reinforcing national identity.
Oh by the way, check this out, this is one of the news in 2025b tzdata:
From https://github.com/eggert/tz/blob/main/asia#L1503.Pretty sure we can find a lot more oddities that are way worse.