What is the ruling on the action of the wife's relative who opened commercial registers in her name and is earning millions without giving her any compensation, despite her demanding it?

Granting a commercial registration to another party is considered a form of guarantee (kafala), and it is not permissible to take compensation for it, because the original principle behind it is to act voluntarily and out of kindness, especially if the ruler prohibits it for a legitimate Sharia-compliant reason.

Therefore, your wife is not permitted to take compensation for allowing her relative to use the commercial registrations under her name. However, that relative should treat her well just as she treated him well, based on Allah's saying: ﴿Is the reward for good [anything] but good?﴾. So, he should maintain ties with her if she is in need and he is wealthy, out of kinship ties, and in reciprocation for her kindness and good deed.