Those are some deep questions. Let me try to explain.

Base uses optimistic rollups, settling on Ethereum. (it's writing the final transactions to Ethereum)

Optimistic rollups are layer 2 (L2) protocols designed to extend the throughput of Ethereum's base layer. They reduce computation on the main Ethereum chain by processing transactions off-chain, offering significant improvements in processing speeds. Unlike other scaling solutions, such as sidechains, optimistic rollups derive security from Mainnet by publishing transaction results on-chain, or plasma chains, which also verify transactions on Ethereum with fraud proofs, but store transaction data elsewhere.

B3 also uses optimistic rollups, but settling on Base. It's also using something called, https://celenium.io/rollup/b3, which is an infrastructure provider for rollups. So you're not able to run a witness node as you'd be able to run one on Hive.

Now, neither B3 or Base are using databases in the traditional sense. Assets don't just exist somewhere in a database, they're simply stored as transaction logs and apps wanting to use that data, need to index the transactions and then create their own databases out of them, i.e. https://ponder.sh. Due to smart contracts aka. logic being onchain, all of the transactions can be used to reconstruct the assets.

It works similarly on Hive, just with the difference that the logic for custom jsons is NOT onchain. For example, if you open a pack in Splinterlands, that opening is processed offchain and then stored within Splinterlands' database. You could broadcast 5x the pack opening, but if in the end only 1 gets processed, simply alone from the blockchain transactions, you wouldn't know which got processed and which didn't.