How did Coinbase monitor BTC transactions?
Your understanding of Bitcoin is correct.  What you may not know is that the "advisable security practice" of using a unique address for each transaction is not followed by many block chain gambling sites, such as SatoshiDice.  Because these sites reuse addresses, transactions sent to those addresses can be flagged or blocked.

Even if this were not the case, block chain gambling may be easy to detect by analyzing transactions.  For example, I believe SatoshiDice always sends back a tiny transaction for each bet it receives, which is an obvious marker.  There may be more subtle indicators too, like the ratio of spends to receives being near the ratio of the amount spent and received.  Also, Coinbase could have access to non-block-chain info like people's Coinbase addressbooks where they write "dice, 1:100 odds" or "weed dealer".