Is purchasing cryptos like purchasing stocks?

Purchasing crypto currencies is completely different from purchasing stocks.

When you purchase a stock, you're buying a share of the underlying company.

When you purchase crypto currencies, you're buying the underlying asset, which you own. The transaction is stored on a public ledger, called Blockchain. No one, but you owns that crypto currency due to the decentralized nature of crypto currencies. This is also why keeping your crypto currency off of exchanges is a good ideas, since you don't own the private keys until you move them to your own wallet.

While purchasing crypto currencies on exchanges is completely fine, keeping them stored there is not considered good practice.