Bitcoin is a currency and no one should ask you for id verification for earning or dealing with a currency. Imagine going to the grocery tomorrow and the grocer asking for your identity to buy a loaf of bread! Wallets do not require any id verifications, but most exchanges do insist on one after a certain amount of transactions.
You have clearly said it yourself that wallets do not require personal information, so why bring exchanges? Wallets and exchanges are for different purposes. Wallet is for *storing* Bitcoin, while exchange is for *trading* or *exchanging* currencies/assets. Because of legal issues, exchanges do require personal information *for trading* [above their limit]. If you use it *only* for storing, there is no need for ID verification, but I absolutely deprecate using exchanges for storing your bitcoins.