Where is my bitcoin balance stored in the blockchain?
The blockchain defines your balance, but it doesn't have a specific field in which your balance is stored.

Yes, it needs to be analyzed to find your balance. However, it doesn't have to be analyzed all over again, just to check for the balance of a new address. Instead, checking the UTXO set (which only has to be created once and then to be updated every time new blocks are received) is sufficient. Furthermore, not the entire UTXO set has to be traversed. There are things like hash maps which make this a lot easier.