When people talk about the price of Bitcoin, they are referring to the current price at which Bitcoin is changing hands. Since Bitcoin is a purely speculative asset, this price is determined by how little sellers are willing to charge and how much buyers are willing to pay. Even then, the price of Bitcoin can vary across exchanges like Coinbase and Binance or currencies because of market inefficiencies
In 2018 Coinbase launched their independant mobile wallet for iOS and Android. The wallet stores the private keys on the user’s device and only they have access to the funds. This brings Coinbase full circle as it started out as a wallet, transitioned to an exchange only (claiming that they are not a wallet) and now they are offering wallet services again.