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.
You can learn more by watching this video.