There are several problems that arise by keeping time across galactic distances, arising from relativity, FTL travel and lack of a unified referential.
How does the "Stardate" in Star Trek work then? Is it the same on Earth and Vulcan, for example? How does the federation keep its value the same across these distances?
According to the Original Series Writer's Guide:
We invented "Stardate" to avoid continually mentioning Star Trek's century (actually, about two hundred years from now), and getting into arguments about whether this or that would have developed by then. Pick any combination of four numbers plus a percentage point, use it as your story's stardate. 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 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.
The Writer's Guide for The Next Generation stated:
A stardate is a five-digit number followed by a decimal point and one more digit. Example: "41254.7." The first two digits of the stardate are always "41." The 4 stands for 24th century, the 1 indicates first season. The additional three leading digits will progress unevenly during the course of the season from 000 to 999. The digit following the decimal point is generally regarded as a day counter.
Deep Space Nine began with 46379.1, corresponding to the sixth season of Star Trek:
Star Trek: Voyager began with stardate 48315.6 (2371), one season after TNG had finished its seventh and final season. As in TNG, the second digit would increase by one every season, while the initial two digits eventually rolled over from 49 to 50, despite the year 2373 still being in the 24th century.
Star Trek: Nemesis was set around stardate 56844.9, which is so far the highest stardate to have been mentioned in the Star Trek canon.
To answer the original question, there was never any rule devised by writers to explain what date was used, so one can only assume they used some sort of GMT (Greenwich Mean Time) across the galaxies, based on a focal point (which may or may not have been the Federation headquarters).
A universal standard of time is not possible.
The only reason a time standard on Earth works in our experience is because we are traveling at speeds too slow and distances too small for the lack of a "true time" to matter. But in the Star Trek universe, this is not the case. The "true time" differential would be significant. And, as a result, it would be impossible to implement.
Your question presupposes what scientists call the Netwon's Stage model of the universe. Essentially, that the universe exists as a static, passive stage upon which all the events of the universe play out with the backdrop of the synchronous passage of time. This model would be necessary in order to have a "consistent" stardate time standard.
However, Einsteins Special Theory of Relativity proved that Newton's Stage model is wrong across large distances — such as the galactic scale distances at play in Star Trek.
To picture how this works, one must first accept the principle that the speed of light is a universal constant. And it doesn't matter where an observer is located in the universe (or how "fast" that observer is traveling) the speed of light remains the same. In order for that to be true, then space and time must be "flexible" and allowed to distort relative to the perceptions of any given observer.
What that means for your question is that if two observers are traveling at different speeds, then the events of the universe play out at different times relative to each observer. In other words, there is no universally consistent concept of a "now" moment. All observers across the universe do not share the same "clock." And, therefore, there can be no universal time "clock" that all observers can agree to. Everyone's clock will be different depending on their history of relative speeds.
