I wouldn’t say its rivalry, but more like nationalism in all three countries. Chinese people feel strongly about making their domestic industries competitive, part of the reason they shut out foreign applications and websites. Tencent, Baidu, Huawei, Lenovo, Xiaomi, ZTE, Meizu are a few of China’s tech companies that are gaining a lot of popularity.
Koreans rely on their top 2 tech giants Samsung and LG, and almost every Korean you see probably owns one of their cellphones or appliances in their home.
Japanese have an insane amount of great tech companies to be proud of. They are the predominant industry for video games with Sony, Nintendo, Capcom, Konami, Taito, Namco, Square Enix etc all being Japanese and in addition they own almost the entire camera market with Olympus, Canon, Nikon, Fujifilm, Pentax, Sony and audio market with Marantz, Denon, JVC, Onkyo, Yamaha, Sony, Pioneer, Panasonic etc
All three countries have things to be proud of. The thing is most of the negative things said about these countries come from netizens and people who have never been to those countries. In reality, when I meet with Japanese/Koreans/Chinese and introduce them to each other (they have to speak english of course), they actually get along quite well when they realize they have more cultural things similar than different. All three countries are known for their hard working Confucian based family oriented cultures.
North Koreans were forcibly separated from South Koreans during the Korean War. To this day, South Koreans cannot visit North Korea to see their relatives or cousins. Taiwanese and mainland Chinese can still freely visit each other.
As for the ‘brotherly bond’ thing its because Korea used to be one nation, and as a small nation people were tighter knit. And its true to this day, you’ll notice South Koreans are still more or less a tightly knit community.
China doesn’t have the same tightly knit community because they are a massive country full of different cultures and ethnicities. Chinese don’t have a brotherly bond with Taiwanese because its doubtful you’ll find within China any brotherly bonds, do Sichuan Chinese care about Hunan? Do Beijing Chinese care about Tibet? Do Southern Chinese care about Northern? probably not. China is much too big and diverse a country to say there’s any “brotherly bonds”.