Unsure about the ”half of the world’s binoculars” claim but they certainly make all or part of quite a few. KK is a very large and established optics OEM based in Japan with extensive global interests and associations. As pointed out above they’re even here in the USA.
The challenge with this thread is it will likely end up like most of these do - often just repeating undocumented claims written in other threads on the web. Some companies like Tract and Maven will gladly share that some or all of their optics products are wholly or partially fabricated by Kamakura. Other, often more established brands are a bit more cagey about it, as some of their models have KK genes and some don’t. So, only true industry insiders know the whole truth. In addition, KK isn‘t the only optics OEM, companies like Meopta and others have also been doing it for decades.
A key point to remember is an OEM builds to the designs, specifications and tolerances agreed to with the purchaser. So, just because two optics are built by the same OEM does not dictate that they are of parallel quality.