Obviously, with no humans living permanently off-planet yet, Earth-style dates and times are the most common system is use, and reasonably so. And it helps that Mars's day is almost the same length as Earth's, so even Martian colonists could use roughly a 24-hour day. (I was always charmed by the idea in the Mars Trilogy of simply stopping the clocks at midnight for a 38-minute "witching hour," a sort of half-hour Mardi Gras every night, which would become a bit of local cultural flavor for people visiting Mars).
But eventually, when substantial numbers of people are living on the Moon and on Mars and on Venus and on ships in between, etc, a 24-hour day and 365-day year will be an increasingly irrelevant timescale. There will probably need to be a "stardate" or some other interplanetary date/time system that is standardized across all planetary bodies, and even later, all solar systems.
A few systems exist today that could work, such as Unix timestamps, but often these measure such small increments (like seconds) that they are better for use by computers than by people. But at some point, a person on Venus will need to say to a person on Titan, "it happened back in February," and I can't imagine that learning the complete calendars of every planetary body will be the most practical option.
Has there been any effort, by NASA or SpaceX or any organization with some kind of practical authority, to establish such a system for tracking long-term times and dates in space? Some interplanetary equivalent of Greenwich Mean Time or Zulu time, for dates as well as hours?