7.62 x54mm caliber the same caliber as 7.62 NATO round?

Remember, cartridge and caliber are two different definitions. The answer is no.

A cartridge titled "7.62" can confusingly be of differing calibers (or diameter). The 7.62 NATO aka 7.62x51 (metric designation) is also known in a US civilian version, 308 Win.

US rifle calibers, including the .308 Win, 30-06 Spr, 300 WM, 300 WSM, etc., are all .308 caliber, i.e., .308"/inch. British .30 caliber rifles (e.g., 300 H&H Magnum) also use .308 bullets.

On the other hand, most European/Russian 7.62 cartridges, including the 7.62x54R use a slightly larger diameter bullet. 7.62x54R is a rimmed Russian cartridge originally used in the 1891 Mosin Nagant rifle, subsequently used in many different Russian/Soviet weapons, and is still used today. Depending upon the manufacturer, the 7.62x54R uses a .310-.312 caliber bullet. If reloading with commercially available US bullets, use .311 bullets for best results. (sources include F.Barnes Cartridges of the World)