The whitepaper just gives a conceptual overview of Bitcoin, but certainly isn't a full specification. Anyway, beyond the blockchain itself, keeping track of transactions and UTXOs is more of an implementation detail than a protocol concern. Various explorers, nodes, and wallets will differ in their approaches, but generally there are a few different mappings between transactions and blocks for different purposes:
Blockchain: block to transaction mapping
First and foremost, the blocks as included in the blockchain include the transactions they confirmed. The complete blockchain is retained in by archival full nodes. As Anonymous already explained, the merkle tree is only stored implicitly in the blockchain per each block simply listing its full set of transactions in order. Full nodes recreate the complete merkle tree on demand by rehashing the block's transactions during block validation as well as when a Merkle proof is requested from them.
Specific interests: Transaction to block mapping
Wallets generally retain copies of the transactions that involve their addresses. These transaction objects should generally keep the block height and the block hash, so that the wallet can deal with chain reorganizations and keep track of confirmation count. I would suspect that some wallet implementations may also keep the Merkle proof around if they e.g. want to be able to proof the existence of funds to other devices.
Bitcoin Core will keep a full transaction index around when configured with the start-up index
UTXO set: Output to transaction/block mapping
Full nodes maintain the ledger of all Bitcoin funds in form of the Unspent Transaction Output (UTXO) set. UTXOs are uniquely identified via their outpoints ([txid:vout]) which means that they inherently reference the corresponding transaction. A UTXO object should generally also include a direct or indirect reference to the block that confirmed the transaction that created the UTXO.
Explorers generally keep more comprehensive databases of transactions, blocks, addresses, and UTXOs.