I'd submit an answer involving jargon like "mempool" doesn't qualify for "ELI5". ELI4?
Mempool or "memory pool". It contains pending transactions that have been publicly broadcasted and are waiting to be mined into the blockchain.
Etherscan indicates that the transaction was in the mempool [0]. My understanding is that this wouldn't show up if it was mined without releasing the transaction to the mempool.
Is this even still possible? Since EIP-1559 miners like ethermine stopped including their own tx for 'free' presumably because you cant pick and choose anymore.
You can still pick and choose, EIP-1559 forces miners to burn part of the fee, which means it's no longer "free" to the miner to include these transactions. In the case of this transaction, Bitfinex also included a very large tip that went to the miner.
Is the mempool logged? Is it possible to tell if it was a mistake or if they skipped the mempool?
Every node operator can log their node's mempool, but the protocol does not ensure that mempools are consistent across nodes.