We can use (Earth based) UTC on the moon, or on the whole universe for that matter. The problem is that on the moon, due to relativistic effects, a UTC second

We can use (Earth based) UTC on the moon, or on the whole universe for that matter. The problem is that on the moon, due to relativistic effects, a UTC second is not really a second, it is slightly longer, and that makes it inconvenient for conducting experiments on the moon. So for moon-based operations, moon time will be used, it can then be converted to UTC for "universal" operations. It is not the first time we use different clocks. In fact, we already have plenty: UT0, UT1, TAI, etc... That's because the Earth rotation is not that great at timekeeping, atomic clocks do better, but we still want our days to be aligned with the sun, so depending on whether you are more interested in the Earth rotation or in precisely counting seconds, you will use different clocks, with UTC being a compromise you can always convert to. GPS time is one such example, it is a few seconds off UTC because it doesn't account for leap seconds. All that is not even accounting for local (earth) time zones.