> But if you have 関係 with somebody in Japan, then that simply means there's some sort of relationship (vendor/customer, former schoolmates, whatever), not necessarily the kind that lets you call in favours.

That's the same as Chinese, though. If you want to say two people you'd use guanxi (”关系不好“ "guanxi bu hao" “have a bad relationship"). Even your above example is used in pretty much the same way guanxi is used. Here's an example from the Baidu entry on guanxi: "小王跟这件事没有关系" "Xiao Wang has no connection to that thing."

From what you write, it sounds like the Japanese use of the word is fairly similar to the Chinese use of it.