Not exactly.
You have no control over your own "money".
- Coinbase can some day decide they want to shut down your account and prevent you from accessing your funds.
- Your account can be hacked and the hacker can take all your "money" that you were storing in your account.
- Coinbase itself can be hacked and you (among all other users) can lose all your funds.

- Educate yourself on how bitcoin works
- Choose a secure wallet and learn about cold storage
- Move your funds from Coinbase to cold storage
Start by reading this: https://bitcoin.org/en/getting-started and this: https://bitcointalk.org/index.php?topic=1631151.0

You don't purchase a key, you generate it randomly. Basically it means selecting a random number in a massive range (bigger than 1 and a little less than 2²⁵⁶). Your wallet does this in the background.