You are using some Bitcoin vocabulary, but you are jumping to conclusions that aren't correct.
Quick vocab session:
- Addresses can be used to receive payments.
- Wallet is a term used for a collection of addresses.
- The sender needs to sign a transaction using a private key
When you receive a Bitcoin transaction, you don't have to sign anything. In a transaction the recipient is passive, similarly to receiving a letter in the mail he doesn't have to do anything at all. Whenever he looks at his mailbox at a later point in time, the letter will wait for him; whenever you check the blockchain your address will have a balance associated with it.
The recipient of a Bitcoin transaction doesn't even have to be online at the time of the transaction. However, he does need to have a Bitcoin address, where the payment may be directed.
The identifier you are asking about seems to be such an address. The good news is, you can have as many addresses as you want.
Wallets are essentially an abstract construct to talk about a collection of addresses. It cannot be discerned from the addresses alone that they belong to the same wallet, so you don't necessarily give up a lot of information when you have several addresses in the same wallet.
However, by analysing interactions between addresses it is possible to guess that they are controlled by the same entity, e.g. guessing which transaction output is the change from a transaction, or funds being spend from different addresses in one transaction.
See What's the difference between a wallet and an address? for more information about that.
Concerning anonymity, I'd suggest having a look at How anonymous are Bitcoin transactions? and the tag anonymity.