How do billionaires keep their stakes in other companies safe during economic crises?

The brokerage company doesn't own the stock. The brokerage facilitates the buying and selling of stock.

The shares you buy in any brokerage are your shares. The brokerage can't just steal people's shares because they're going broke.

The more interesting question in this case would be where does he (or where do they) keep their cash. The cash actually might be at risk.

Stock issue, transfer, and holding is a system of accounting. A stock-transfer-agent keeps an accounting of a company's stock. If the transferred stock is not held in the investor's name then there is a second accounting by a stock-broker who is holding the stock in their name but accounted on their books to their customers.

The real risk is the company invested-in itself. A company with a loss of revenue and a large amount of debt is at risk of bankruptcy. Then in a re-organization the shareholders are often wiped out with the bondholders becoming the new shareholders.

Public companies use stock transfer agents to keep track of who owns their shares.

Direct registration

Under direct registration, when you buy a stock, the stock is not held in your broker's name. It is held in your name and your name appears on the stock transfer agent's list. In this scenario, there is no "custody" as the stock is held directly under the investor's name. The brokerage firm could disappear tomorrow and the investor would still have his stock intact.

Street name

Alternatively, the investor's securities could be held in someone else's name (a "custodian" or "nominee". See: street name securities). As you alluded to in your question, this scenario involves risk. The risk is mitigated to some extent by SEC regulations (e.g. requiring brokerage firms to segregate client assets from firm assets) and compliance monitoring by FINRA. For small investors, the Securities Investor Protection Corporation (SIPC) provides the insurance against loss should the nominee firm collapse.

Conclusion: If the shares owned by Berkshire Hathaway are held in its own name, there is no risk of loss from the collapse of any nominee firms.