You can use your Coinbase wallet in a variety of ways, including having it build a cart and payment address for you for each transaction or simply using a single address for receiving multiple ongoing payments. The address will be permanently assigned to your Coinbase account once you create it.
That said, I don't think there is a way to get the private keys for these addresses so you can use them on your own wallet software. Not that you would want to, given Coinbase is currently holding (and aware) of said addresses keys.
Here's some Python code for creating addresses on Coinbase, if anyone is interested in how it works: https://github.com/StackMonkey/utter-va/blob/master/webapp/l...
That said, I don't think there is a way to get the private keys for these addresses so you can use them on your own wallet software. Not that you would want to, given Coinbase is currently holding (and aware) of said addresses keys.