However, the term "crypto wallet" makes people think their Bitcoins, ETH, or any other cryptocurrency is stored within the software or device. This is far from the truth.
Most cryptocurrencies are stored on the blockchain. The crypto wallet is mostly a user interface allowing you to interact with the data stored there. So, Ethereum wallet enables you to see the balance in your account and to perform transactions with it, although the account is actually stored on the Ethereum blockchain.
What is crucial is the private key. It is the cryptographic information that allows you to prove ownership over the account and to execute transactions. You can share your public address but never reveal your private key.
So, when you receive any cryptocurrency, coins do not physically get into your phone or hard wallet. They are recorded by the blockchain in the corresponding account or ledger.
The wallet just provides access to the blockchain.
That is also why the loss of the wallet app does not automatically mean the loss of the crypto. If you remember the recovery phrase or the private keys, another wallet app that is compatible with your crypto wallet can give you access to the account.
The principle on which hardware wallets operate is the same as well. The aim is to keep the private keys safe rather than keep blockchain’s coins in the device.
That is also the reason why your recovery phrase is so valuable. Once someone gets his hands on the relevant private keys or recovery phrase, he will be able to control the assets.
That is why the term “wallet” can be quite misleading because the wallet does not contain any coins like a real wallet contains cash.
It is the blockchain that keeps the records of the assets while the wallet gives you control over them.