Specifically, 7.62 mm seems to have been quite popular for small arms, as well as 20mm, 37mm (pretty much everyone who ever fielded a cannon), 57mm (UK, USSR, Japan) 75mm (probably the most popular modern cannon caliber), 76.2mm (USA, USSR, UK), 105mm( UK, USA, Germany). I am almost definitely missing a few nations that used those calibers.
Try /r/forgottenweapons. That guy knows his stuff.
Root cause: "Why would a single nation have such a limited set of calibres?
Standardized ammunition leads to inter compatibility of weapons. This means you can introduce the new weapon in an evolutionary way rather than a "big bang" revolution.
When you introduce a new firearm - or modify an existing one - you want to be able to use it. If you have to re-tool an entire manufacturing pipeline, that makes things harder.
An army will have an existing inventory of ammunition. That inventory will be supported by a manufacturing base, a logistics system and a set of known safety processes.
On the battlefield, in contact, if your oppo screams "AMMO!", you want to give him a mag he can actually use. "Sorry mate, mine's 556" will get you both killed.
With these constraints - manufacturing, logistics, handling, familiarity, interoperability - the real question should be "Why were there so many different calibres of small arms?"
So: why did so many different nations adopt the same calibres? Answer: Most nations don't design and build their own weapons. And, in the 20th C, most leaned on one of the main power axes for weapons.
All the reasons above apply to each bloc because of network effects.
History of War has many good articles discussing the decisions involved in delivering new weapons.
This article describes some of challenges of a calibre change - the 7.62 SLR to the 5.56 SA80.
TL;DR - Systems Engineering principle of "Change as few things as you can" coupled with "Use what you already have.", followed by the Cold War two-axis alliance structure.