[SteveS]

Many British imperial units remain in customary use in the US, but apparently liquid volume units were not fully standardized when the US and the British Empire parted ways, so when the standards settled, the US had four quart gallons and the British Empire had five quart gallons. I don't know whether US gallons have ever been a customary unit anywhere else.

So it's fair to say that the customary units in the US are "Imperial, with US liquid units". But because the rest of the former British Empire has done away with the old Imperial units, it probably avoids confusion most to just call the units in customary US use "US units", even if they originated in the British Empire (again except for liquid units).

As for the units used in the Third Imperium, I think Traveller history has made metric the standard, with the addition of the dton. If the history defines how that came about, I haven't seen it. If I had to make up an explanation, I'd give a history like this:
- Vilani units were dominant before the encounter with the Terrans.
- Throughout the Interstellar Wars there was* a hodgepodge of Vilani, metric, and US units in use (with Vilani units dominant in anything Vilani or reverse engineered by the Solomani).
- The dton caught on even with Vilani, because of the practical appeal of a unit that was so close to one one-hundredth of the minimum size of a jump-capable ship, but they refused to acknowledge that the unit was adopted from the rimward barbarians.
- Early in the Rule of Man, an administrator campaigned for standardization on metric. The campaign was most successful on worlds with strong anti-Vilani sentiments and their commercial dependents, and on worlds depopulated by plagues, but largely rejected in the old Vilani core. The Standard Day of 86400 seconds caught on in commercial timekeeping.
- During the Long Night, worlds generally stuck with whatever was popular by the end of the Rule of Man. Lots of locally customized units developed, particularly in the measurement of time. The 365 day Standard Year (dropping the complicated Gregorian leap year rules) caught on in a few places where enough trade carried on for multi-world timekeeping to remain useful.
- Sylea was a world that had adopted metric before the Long Night, and part of a region where the Standard Day and Standard Year persisted for multi-world timekeeping. It carried metric, the Standard Day, Standard Year, and dton into use as Third Imperium standards.

Some other standards are mentioned in canon, particularly in the area of timekeeping. For examples, the Zhodani have their years and olympiads, the Aslan have their time units, Hivers have their base-six units, and the Vargr have their typical disunity.

* Should this be "was" because "hodgepodge" is singular, or "were" because "units" is plural? Hmmm.