There is no canon answer.
It seems reasonable to speculate that JK had intended simply to convey at that early juncture at one or both of two things:
- Wizards use very different timepieces than the muggle world does.
- Wizards have significantly more information displayed on their watches, including stellar states.
It may also be possible that Dumbledore, being Dumbledore, had a unique timepiece that he himself used with distinction from everyone else.
The way that wizards keep time versus muggles was likely not more than an early idea that did not transform into something more significant, and was simply dropped or forgotten as the story developed in lieu of your standard West European time system. This kept non-essential distractions out of the already increasingly lengthy plot line.
I will make note here that the Western 24-hour (mod 12) timekeeping scheme is not the only common one in the world today: there are around a dozen other systems still in use around the globe.
Ultimately, however, without more information from Rowling herself there is only speculation.