All the data is contained in the blockchain. But there is no way to “undo” a transaction, that’s the essence of the security features.
It’s a kind of “going forward” only type of implementation.
And anyone can get a wallet, it’s afaik completely anonymous.
Empty wallets are anonymous along with as many private / public key pairs as you care to generate. But those don't hold any coins until you transact some into your wallet onto one of your keys. To do that you have to send someone your public key and have them transfer some coin onto it. Basically they have to pay you. Unless you know some generous people, you are going to have to bring some cash somewhere and find a person to physically transfer the coins on the spot.
Or you can send money to an exchange and they will buy the bitcoin from a seller at the same exchange (like the stock exchange). The bitcoin will then be owned by the exchange (their private and public key) and you have to then ask them to transfer it to your public key. Then it will be yours. The trick is that a lot of exchanges want ID.