The US inch was originally 25.39978 mm, and the British Imperial inch, 25.39998 mm. I understand that a Scandinavian company that manufactured extremely high precision measuring apparatus, decided it was simpler to use 25.4 mm exactly, which was more than accurate enough for precision engineering purposes. The US and the UK eventually decided to change their official standards to follow suit.
The difference between the US and the UK fluid ounces is a consequence of them being defined as the volume of an ounce of water at different temperatures, on account of the different mean ambient temperatures at their respective standardising authorities back in the days before air conditioning. Water expands with increasing temperature, so an ounce of water at the higher US standard temperature occupies a greater volume than an ounce of water at the cooler English standard temperature. I have a bottle of ink that gives its contents in both units, 2 Imp.fl oz, 1.9 US fl. oz.
The US Gallon is based on the Winchester wine gallon used by the original colonists. Until the UK established Imperial standards in mid-Victorian times, no doubt in part due to expansion of the railway network, local weights and measures were in use in different parts of the UK.
I used to have a Victorian school arithmetic book whose preface mentioned that it was a revised edition which had deleted the exercises in converting between local weights and measures, as the establishment of national standards had made them unnecessary. The standard foot, 2ft and yard mounted in the wall at the North side of Trafalgar Square date from this time.
Many things that we consider to be American are things that originated here. The Americans have continued using them, while we have changed our usage.