> How exactly does the Government of Japan apply pressure to a public company to merge with another failing public company?

Structure doesn't matter. Culturally government cooperates with companies through "asking" (or pressuring if you like) as opposed to western approach where companies can (and will) do as they please within law/regulatory frameworks. Opposite works as well - companies can ask government and pretty much expect result.

Most of it stems from collective culture and family values and taken as something quite important.