I think this comes from two things: there is a lot of company-loyalty, from Japanese drivers especially. Guys like Sato are Honda through and through, while Kobayashi and Nakajima are Toyota men. There doesn't seem to be much back and forth on this front.
Second, the more high-profile the athlete, the more value he or she has for sponsors and suppliers. That tends to come with exclusivity arrangements. If even small-time suppliers for things like team clothing can tell these people what to wear, I'm not sure it's so far fetched to think that a Toyota or a Honda wouldn't be too keen on their star driver also appearing for the other side. Consider also that Toyota and Honda rule the Japanese car market, with less than 10% of Japanese sales coming from outside the country. These two are more directly involved each other's business than, say, a Porsche and Mercedes (Hülkenberg's outing for both in 2015 comes to mind).
Japanese companies have a particular relationship with Japanese drivers, not the other way round. Rather like Red Bull scouts from the companies will follow karting etc and pick out promising talents, smoothing their way through junior formula primarily with money. It isn’t as developed as RB with little assistance on media handling etc and IMO the scouts are not particularly good at what they do. IMO they’ve done the Japanese racing scene quite a bit of harm as a result.
But it is true that a driver like Kobayashi who has had their entire career paid for by Toyota until he got to Sauber would firmly be expected to race for them and be looked after as a result. Sato was a little bit more complicated.
This doesn’t apply to foreign drivers although canny individuals like Jenson can use their Honda connections to race elsewhere in the empire.
Previously I have said that Honda do not want to work with Alonso again and that is still my understanding. But if he hadn’t fallen out with them then I still think it would be difficult for him to sign at the same time as racing for Toyota. Not just because Toyota is Toyota but it means paying a large sum of money to a driver who may become unavailable to you because of an accident they’ve had in a completely separate sport contracted to a different company.