Universal Universe Time/Analysis
This trope gets even worse when you start considering General Relativity, which states that every location sees time pass at a different rate, depending on how fast it's moving and how deep it is in a gravity well. So even if the faraway colony of Galvenon Eta Prime did decide to ignore their own day/night cycle and use Earth time, there would be no way to keep the clocks in sync without a precise knowledge of the relative velocity and the depths of the gravity wells involved. And further General Relativity also comes with the consequence that there is no universal time, there isn't even a universal way to order two events unless one might have caused the other, for any two events you see as being simultaneous someone else can see either one of them happening first depending on relative position/velocity/etc. The question "What is happening right now on Alpha Centuri?" doesn't really mean much unless you are actually ON Alpha Centuri, because for anyone a significant distance from you when "Now" is isn't well defined. The best definition of "Now" for some place far away is pretty much any time between the light you can see from that location at the moment you ask the question, and when light from that moment will reach them from you. All of this falls out of "Assume everybody sees light as going the same speed, no matter how fast they go relative to each other." Seriously, that gets you all of Special Relativity.
Even on Earth, the non-uniformities of the Earth's rotation mean we get weird things like leap seconds, and at the nanosecond level relativity requires clocks to be corrected for altitude. Here is a list of various timescales that have been used over the years, many of which have become obsolete as more precise clocks revealed problems like irregular rotation and relativity.
TCB (Barycentric Coordinate Time, with a French acronym) can be used as a time reference for anything that happens in the solar system. So can TDB (Barycentric Dynamical Time), which is TCB rescaled to pass at the same rate as TT (Terrestrial Time), averaged over a year, but at any given moment differs from TT by up to a couple of milliseconds due to the eccentricity of the Earth's orbit. TCB, TDB and TT are ideal times, which no clock actually provides – TAI (International Atomic Time) tries to track TT (except for being just over half a minute behind for historical reasons), but the difference (TT - TAI) has fluctuated by a few tens of microseconds due to technical difficulties. UTC (Coordinated Universal Time, and no, that acronym doesn't work in any language) is the one with the leap seconds – since 1972 it has been a whole number of seconds behind TAI, to stay within 0.9 seconds of UT1, which is based on the Earth's rotation. It is currently over a minute behind TT, and this gap continues to grow as the Earth's rotation slows down.
Note that the above paragraph only deals with one solar system, with one inhabited planet. So clearly, timekeeping in an interstellar civilisation will make many people's brains hurt.