The wristwatch on the desktop is not clickable. It displays the current time according to your computer’s system. (If you reset your computer’s clock, it resets the time on the watch, and the day to night changes across the site.)
The marks around the face are that of a compass, not a clock. with 16 marks rather than 12… even though it is numbered with the four traditional quarter hours: 12-3-6-9.
During Peeves‘ visit, the watch goes wacky.
Then small dots appear around the edges in place of the hour markings and 12 hands appear one at a time around the face.
The dots are slightly different sizes and colors (planets?), leading me to believe it is a representation of Dumbledore’s watch, which is described as having “12 hands but no numbers; instead, little planets move along the edge“.
After the doxy flies through, the watch face emits a flash of white light and returns to normal.
JKR.com: The Watch