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I can't find a mining reward transaction hash for a given block. I can see the block hash, all the transactions inside the block, etc. But the mining reward transaction is not listed among the details.

Does a mining reward transaction have a hash? Does it have inputs and outputs? I tried different block explorer sites and JavaScript web3 functions. What kind of transaction is a mining reward? How is it implemented?

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There are no inputs and outputs in Ethereum, just state changes and balances. Therefore, mining rewards don't have a transaction hash since they are not a transaction.

Each block contains the miner's address, and when the block is published, the miner just gets 5 more Ether added to their balance, without the need for a transaction.

See this question for more about UTXOs vs Balances

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Here you can find the code increasing miner's balance with reward amount without any TX being created: https://github.com/ethereum/go-ethereum/blob/v1.9.10/consensus/ethash/consensus.go#L638

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