Is there a way to accept bitcoin payments and verify the identity of the people who paid without asking for any other information? I have a website and I am considering supporting bitcoins, but I would like to do this without asking the users to log in nor give me their email or details before buying.

Is there a way to accept bitcoin payments and verify the identity of the people who paid without asking for any other information?

I would like the buyers to claim their product after they pay, using the transaction id, the confirmation number, the address... or something like this, so I do not have to ask for any other information before they buy, but I can be sure that I deliver the product to the right person.

That is, you can pay directly using my btc address (that will be different for each transaction) and you do not have to enter anything else. 
After the transaction is confirmed, you can come back to the website and you can claim the product entering a "code". (Only then you'll be asked for the email or more details).

Can I use the pair of transaction addresses as this code? Is this secret or other people will be able to claim they paid?

Thank you very much!!
You could have the customer send a message signed with the Bitcoin address used. This can be done with most Bitcoin clients or with most of the online wallets. This proves they have access to the private key for that address and are the owner. 

Where they send the email from is not important as long as you can verify the message. 

https://blockchain.info/wallet/features


  Sign Message - Sign a message to prove you own a bitcoin adddress.


https://bitcoinarmory.com/start-page/what-is-armory/features/


  Message signing with your Bitcoin addresses:   sign messages using the private keys of your bitcoin addresses, that can then be verified by someone else.  For instance, you pay someone 200 BTC for merchandise to be shipped to you.  You send the seller your postal address, but someone intercepts the message and replaces it with their own address!  This is avoidable by using signed messages:  the seller sees the money came from address X, so you send them a signature block, which tells them your postal address, and sign it with address X.  If someone tries to change the address, the signature will become invalid!