When we hear the term "wallet," its hard not to conceptualize a container, or, at the very least, an account of sorts, much like the online accounts held by our banking institutions. But, in order to have accounts such as these, it would require a centralized entity to maintain and service the accounts.
That's why the blockchain or web wallet needs to be reimagined to something closer to what it really is-rather like a proof of shareholding account that reads the entire script of transactions (depending on validation mechanism) which shows how many shares each and every address is authorized to use. Every node on the chain can read the script and has a duty to participate in validation of accuracy of the script of transactions.
In other words, cryptocurrencies don't move across the chain to come sit in our wallet as if it were a vault. Rather the total supply of coins or tokens sit in a supply vault together, and all of our address wallets hold the actively changing ledger which shows who is entitled to what is inside this vault. Our address is the "keys" to the vault which holds our share. And to be even more specific, this vault isn't a vault at all; it's just a master number on the blockchain.
In other words, we never move crypto in and out of our wallets, or around the blockchain. Rather, our wallets are like our I.D. cards which hold an active electronic balance of what we own out of the giant "pile" of coins. And that electronic balance is accurately achieved and maintained by holding the entire validated, immutable history of the blockchain. For clarification, full node wallets hold the entire blockchain transaction history, while light wallets interact with the full nodes for data and validation. Full nodes operate by supplying computing power to the blockchain, and light wallets operate by connecting to the internet and interacting with the servers which have the full copy of the blockchain. For blockchains such as Ethereum, light wallets sign smart contracts.
I hope this shed some light on the subject; if you have more to add please comment!
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