The site https://corporatefinanceinstitute.com/resources/careers/companies/vanguard-group/ had this to say:

Ownership Structure of Vanguard

Unlike other investment firms, Vanguard is structured as a client-owned company with no outside owners seeking to profit from the company’s investments. The company is owned by its funds, which are then owned by the shareholders. This makes shareholders the real owners of the company, and there are no outside investors apart from the shareholders. This structure not only prevents conflicts of interest but also lowers the amount of expenses charged by its funds. Vanguard managed to reduce the average expense ratio for its funds from 0.89% in 1975 to 0.12% in 2016.

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So I think each ‘Fund’ is set up as its own business entity which is then owned by its investors, who become the shareholders. Exactly how each business entity can then be the owner of another larger entity, the Vanguard Group as a whole, I’m not clear on. But I think the important factor is that each fund is a distinct entity from the others, and it is owned by its fund holders.