The blockchain doesn't know your total possession. You know that addresses A and B belong to you, but the rest of the world doesn't know they belong to the same person.

However, Bitcoin transactions are allowed to have multiple inputs. So you can create a transaction with two inputs, one from each of your two previous transactions. This transaction will include two signatures, one for A and one for B, thus proving to the world that the respective owners of A and B (whether the same person or different people) have authorized their coins to be spent in this way.

Your wallet client software will take care of this for you automatically. So from your perspective, all you have to think about is that you own 2 BTC. You don't really have to pay attention to which addresses they live in; when you want to spend coins, your client software will choose appropriate inputs from among all your addresses.

It's not really correct to think of a Bitcoin address like an email address that identifies a particular person. It's a token that's associated with a particular private key, such that coins sent to the address can only be spent by whoever holds the private key. But there doesn't have to be a one-to-one correspondence between addresses and people.