Wallets are not an essential component of a blockchain. Blockchains are simply a linked-list of data, with each new block of data ensuring the integrity of its previous block. That's it.

If you want to verify that a particular operation on the data was done by the person who is allowed to do it, you need some sort of public key crypto, right? If someone steals your private key, they can pretend to be you on the blockchain. Whether that private key is called a wallet or something else.

There was some buzz recently about Wal-Mary using blockchain to track pork through the supply chain. I thought it would be cool if they e.g. created tokens by hashing a pig's DNA.

That's not really true. We had hash chained data structures for ages before BTC came out. Nobody retroactively called these "blockchains".