When you own stocks or bonds in a brokerage account, they are held in "street name". IOW, the name that appears on the security is that of the broker but you retain ownership rights. This facilitates speedy trading and reduces trading costs.
SIPC protection is $500,000, which includes a $250,000 limit for cash. SIPC protects the custody function of the broker dealer so if your broker goes out of business, your securities are safe.
SIPC does not protect you from your securities dropping in price nor does it protect you from receiving bad advice from a broker.