Jim Keenan offered:
No good reason. 7.62 mm converts to .300" which is the bore diameter of the 7.62 NATO, .30-'06, etc. .308" is the groove diameter. Why the Russians chose a bit larger bullet and why they called it 7.62mm (instead of, say, 7.7mm/.303"), I have no idea and I doubt anyone does at this stage.
I do happen to know why .30 was chosen by the U.S., though. Around 1890, when the Army was experimenting with reducing the standard rifle caliber from .45 to something smaller, an officer at Frankford Arsenal decided to use .30 (.308), thus setting the standard U.S. rifle caliber for the next 75 years. When asked why he chose .30, he replied that "it seemed like a nice round number."
Thus are decisions made.
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Jim:
Now that you mention it, it does seem like a nice, round number, however chambering a U.S. 30 caliber round in a Russian rifle, assuming that the bolt would close, isn't likely to present a problem. The other way round, again assuming that the bolt would close, you have a bullet diameter perhaps .003 - .004 oversize, which could raise pressures significantly, or so it seems. I had thought that, re how decisions were made, that one closed their eyes, turned around several times, and then tried to pin the tail on a donkey.
The old FA Match bullet, the one that U.S. 30 caliber Match Ammunition was loaded with, nominally 173 grains, a few I've weighed scaled 175, no big thing. These 30 caliber projectiles miked .308 - .309" dia, and I once came upon some Norma Match Bullets, I think they were Norma, that miked .309"dia. Had no trouble firing them in a Garand.
That old adage about watching what you stick in that chamber comes to mind, though in this case, watch the bullet diameter. Anyhow, I started out curious, and remain curious.