I'd say "stake" is a general generic term for ownership by an individual or group of individuals in a corporate or other business entity; or a game or other enterprise. "Stock" is a very specific type of financial instrument, controlled in the US by the Securities and Exchange Commission. If a company has 100,000 shares of stock outstanding, and person A and person B each own 100 shares, you can say they have equal stakes in the company.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.