It is a curious thing - I think for the foreseeable future, we'd likely still track age via Earth years for the sake of avoiding this kind of thing. I wonder if/when we do colonize, how long would it take for the Martians to actually switch. How much would our cultures have drifted by that point?
And a related shower thought - until relatively recently (late 1960s), any definition we used to track time has been crazy localized to our planet, compared to other things in math and science where many of those concepts would still work the same elsewhere
A lot of science fiction writers try to address the problem of time when humanity becomes a space-faring race. Star Trek has the idea of a "Stardate" and instructed the script writers to just fucking make it up,
For example, 1313.5 is twelve o'clock noon of one day and 1314.5 would be noon of the next day. Each percentage point (sic) is roughly equivalent to one-tenth of one day. The progression of stardates in your script should remain constant but don't worry about whether or not there is a progression from other scripts. Stardates are a mathematical formula which varies depending on location in the galaxy, velocity of travel, and other factors, can vary widely from episode to episode."
Meanwhile, Asimov in the Robots / Foundation universe, everyone still uses the idea of a 365-day / 24-hour day "year", even if no one remembers Earth (except a R. Daneel Olivaw and a few others).
And Kim Stanley Robinson in his Mars trilogy does what OP notes -- Martian years are longer, and the societies diverge pretty rapidly, within a generation, for a whole host of reasons.
When I was last working on sci-fi writing (for a game I was running), I came to the conclusion that most places that are in contact with galactic society (in a setting with FTL travel and communication) would probably have one calendar that is standardized for official use, and at least one that is local to the planet. Systems with multiple inhabited worlds might have a system-wide calendar in addition to the individual calendars.
On the other hand, people living on ships, space stations, or remote outposts might track everything in standard time.
Dealing with characters who negotiate contracts across interstellar distances and frequently move between ships and planets made keeping track of things like "what time is it planetside if we rest for 8 standard hours on the ship?" and "if we agree that this contract in another system begins in 30 standard days, but we're occupied here for another local week, how much time do we have for travel?" enough of an annoyance that I went looking for a program that could automatically track these things, but I don't remember if I ever found one.
Even our sense of time comes from them. The time it takes the Earth to spin on its axis; the Earth to go once around the Sun. On Jupiter, you'd be celebrating your first birthday. It's hard to feel we matter out here, isn't it? The distances are so vast—it's hard to believe that we can make a difference.