This is more or less how anime gets made. The funding is provided by a group called a production committee in exchange for IP rights. Usually the production committee is a shell company formed out of representatives of the investment wings of various large companies, but a production committee can be a single very rich person in theory.

The production committee contracts with the studio to have a show made (with agreed-upon constraints, although studios typically have a whole lot of creative freedom). Sometimes the idea will be original to the studio, & in that case, the studio shops the design documents (and sometimes a whole series bible) around to various potential investors, often for years, until they get enough interested parties to form a production committee and secure investment; other times, the production committee forms around an idea or an existing franchise (often, a manga, light novel, visual novel, or mobile game will be considered for an anime tie-in as part of its marketing strategy & a production committee will form around it, spearheaded by the marketing agency in charge of the original work), & the production committee shops the idea around to different studios.

If the studio didn’t find investors or the production committee didn’t find a willing studio, the show wouldn’t be made. In either of these cases, a single investor with a lot of cash can make the deal a lot juicier, drastically increasing the likelihood of a show being made.

A billionaire still might not be able to personally fund an entire series — at least, not without really substantial personal risk. Even though key animators are paid below a living wage, tweeners even less (their labor outsourced to South Korea), & almost everyone in anime production working large amounts of unpaid overtime constantly, making an anime is still extremely expensive (in part because modern labor-saving tech common in western animation is not well integrated into the process — while everybody moved to digital coloring around the turn of the century, only CG artists & web-gen animators actually draw on tablets, with most frames drawn by hand on paper & scanned still, & tweening is performed manually by South Korean studios instead of being done automatically by vector animation applications like Flash). Individual episodes cost many thousands of dollars.