Vanguard is listed as being "privately owned" because it's not a public company. It is owned by its own mutual funds. If you really wanted to dig down into it, Vanguard is owned by its millions of customers who are real people.

Vanguard is listed as the major owner of Black Rock, because its mutual funds buy stock in every company. Vanguard is listed as a major owner of every company in the USA (and to a lesser extent internationally), because many people buy their mutual funds in their 401K plans and IRAs.

I don't see anything nefarious going on. I'd recommend reading up more on how mutual funds work.