In the UFP and other known space travelling societies the old time systems of home planets and colony planets are still used. Earth people still use various Earth calendars with years, months, weeks and days to keep the date, and Earth people still use time keeping units like seconds, minutes, and hours.
So people who live on a colony planet or space station or moon base equivalent would use both the local time and date keeping methods to keep track of changes in local temperature and light, etc., and also the date and time keeping system of the home planet of their species to keep track of their ages and celebrate religious or secular holidays, etc., etc.
And no doubt they have simple computer clocks in their various gadgets to keep track of the time and date in both systems for whenever they need to know it.
And when people from various different species get together there are several different sets of date and time keeping systems involved. But converting between such systems should be no problem for the computerized time keeping systems they would have available.
And apparently stardates are used as an interspecies date keeping and time keeping method. Presumably people convert their units into stardates to communicate with aliens and when they are told stardates translate them into their units.
Memory Alpha says:
Stardate systems were used in certain cultures as early as the 2150s, when the United Earth government worked with Gregorian calendar dates. In 2154, Degra, a Xindi-Primate, sent a coded message to Enterprise NX-01 containing a stardate for when Enterprise should rendezvous with Degra's ship. T'Pol knew that it was three days in the future, indicating that Vulcans also had an understanding of stardates at that time. (ENT: "Damage")
The script for "Damage" says:
T'POL: I discovered a set of coordinates embedded in the document.
ARCHER: It's not a star system.
T'POL: There was another embedded number. A stardate, three days from now.
ARCHER: How far away are the coordinates?
T'POL: Four light years. We'll need at least warp three to make it in time. It's unlikely that Degra will wait beyond three days.
Since stardates are not used by Earth people in other Enterprise episodes, the stardate mentioned in this episode should be in a system used by various space traveling species in interspecies communication. This stardate system should have been around for years, decades, or centuries before the first episode of Enterprise, and might have been created by various non human societies before humans ever had warp drive.
If that stardate system was created by non humans with no input from humans, stardates would not be based on Earth time units. And if that stardate system was the system that came to be used by Starfleet in the era of TOS and the era of TNG, people who suppose that stardates are based on Earth time units would be wrong.