Yes. Every validator in Stellar has a copy of the complete ledger. However, different validators may be authoritative for different types of token. Say bank_A runs a validator and issues digital dollars on Stellar, while bank_B runs a validator and issues digital euros on Stellar. Each validator will store both banks' token holdings and prevent double spends. However bank_A should offer to redeem its digital dollars for real currency only when the redemption transaction commits on its own validator, and similarly for bank_B.
Running a validator protects a token issuer against double redemptions, as might happen in a mining-based blockchain where anonymous miners fork the blockchain and thus create twice as many tokens. That's fine for pure crypto tokens, where you can create Ethereum [classic] or Bitcoin cash out of thin air. But if you were using colored coins or ERC20 tokens to represent claims on bank deposits, these forks would be a problem.