Date: tracked and recorded. The game logs are based on what happens each game day (rather than each session).
Calendar: homebrew, 9 months of 40 days each = 360 days, no variance - not even a second per decade - because those who built the world designed it that way.
Holidays: noted in the players' write-up, seldom bothered with much in game. I could - and probably should - design different holidays and celebrations for each culture and-or pantheon, but life's too short.
Birthdays: players track these (roll a d-9 for month then a d-40 for day) and sometimes make use of them in game.
Moon phases: they are noted but not tracked closely unless it's relevant, and it only ever becomes relevant if a) lycanthropy is involved, b) if it's a clear night and someone is trying to see in the dark, or c) if someone actually tries to go to a moon. Tracking moon phases in my current game is a mess as one of the moons goes "backward" through the sky (reverse orbit) and the two moons are on vastly different rotations.
Astronomy: somethng else I really should put some time to, figuring out some constellations and star charts; other planets, etc. I know some DMs get all kinds of story ideas out of such things; I guess I ain't one of 'em.
Lan-"and to confuse things, the length of a month is equal to the cycle of neither moon"-efan