What is Wallet?
A wallet stores the private keys that authorise transactions and prove ownership of an address. It holds keys, not coins: balances live on the chain and the wallet signs for them.
In practice
Self-custodial wallets keep keys on the user’s device, so recovery is the user’s responsibility. Custodial services hold keys for you, trading control for convenience. Smart-contract wallets sit between the two and can add recovery and limits. On Dapping a connected wallet is one of several ways to prove an identity claim.