How exactly would anyone know from where I got a given "bit of Bitcoin", or who I send it to?
All Bitcoin transactions are public information, and each transaction is uniquely identifiable via its txid. Each piece of bitcoin (called a transaction output) is uniquely identifiable as well by means of the transaction that created it, and the index of the output on the transaction. This makes the "outpoint" txid:vout the unique identifier of a UTXO.

So, what your trading partner learns is your name via the bank transfer, the address you gave them, and the transaction output they created for you. That alone is not very informative as you noted, but it can become useful in combination with large swaths of other information. For example, if multiple of your counterparties on Bisq collected the names of whom they trade with, they could cluster your addresses and unspents to the same name. They can also track your on-chain activity: when you spend the funds, they will assume that any other funds used in the same transaction are also owned by you, clustering the other addresses to you. When you send to a recipient and receive the remaining funds back as a change output, they may be able to guess which of the two outputs went to a recipient and which went back to you. In that case, they will be able to probabilistically associate your next payment to you as well. They may have information on the address that you sent to, because you paid into a KYC'ed entity like a payment provider or Bitcoin exchange. In that case, they learn more about the purpose of your payment, or potentially could even look up other account activity if you deposited into an account.

So, basically your bitcoin activity is generally private, in that no personally identifiable information is directly collected in the bitcoin network, yet it is also inherently pseudonymously associated with you which reduces your anonymity set depending on how traceable your activity is. However, since the blockchain is a permanent record, anyone sufficiently motivated and well-connected may be able to fill in the transaction graph to trace back step by step to you.

If you want to read more about privacy in Bitcoin, I would recommend this excellent treatise on the Bitcoin wiki: Privacy