Every time a wallet is created a private key is generated with it. Different wallets may support different tokens.
A wallet can refer to a group of crypto's (on a platform) and the private key is used to verify transactions. When using online platforms you will usually see there is only one public key for a particular crypto.
Public keys however can be generated and are not necessarily unique like the private key is. When using a hardware wallet like a Ledger device, your private keys can generate different public addresses that lead to the same wallet.
I hope this clarifies things a bit!