Bitcoin address does not show transaction/balance from Coinbase purchase I purchased a bitcoin using a Coinbase account. When I try to verify that transaction to the address, I can't see any activity or balance for the bitcoin address that Coinbase shows as mine. Coinbase, however, still shows my purchase activity and balance correctly on my account.

Has this something to do with the way Coinbase or such wallet services work? Or is there something wrong? Do I actually have the bitcoin or not?
The coins you purchase are stored in an address owned by Coinbase.  This allows them to buy and sell coins using their hot wallet as well as allowing them to move excess coins to cold storage.  It also allows them to process send requests between Coinbase users using off-blockchain transactions.

The coins won't show up in an address owned by you until you send the coins to a non-Coinbase address.  When you do send the coins, two transactions will be generated by Coinbase.  The first transaction moves the coins from an address controlled by Coinbase to a new address controlled by you.  The second transaction then moves the coins from this address to the address you specified in the send request.

You can send coins to any of the addresses shown in your Coinbase wallet.  Or you can generate a new address at any time to receive deposits.

Since Coinbase keeps just a percentage of the coins in their hot wallet, this can lead to a delay in sending coins if they need to move coins from cold storage back to their hot wallet.