Pretty much every cryptocurrency (bitcoin, etc, etc) wallet does not ask for your information because that's part of the idea of cryptocurrency. I recommend downloading the extremely user friendly wallet called Exodus. The magic of modern crypto wallets is that they persist even if you destroy the computer you used to download and host the wallet software. If that happens you simply download the software again and use the 24-word passphrase to reload your wallet and all the money. As long as you keep that 24-word passphrase safe from theft, loss, hackers, etc. then your money is safe and persistent. For extra-security I recommend buying a hardware wallet from an approved manufacture and NOT eBay or other non-approved re-sellers
Next, most exchanges don't ask you to verify your identity but they technically know who you are when you transfer money to the exchange via a bank transfer. If you are concerned about that then you should look into peer-to-peer exchanges, basically a system that connects you people that want to buy or sell crypto. The way it works (as the buyer) is that you connect with a seller and the sellers moves the crypto to a "escrow" wallet that holds the crypto until you send the seller money. You can do this is any number of ways from direct bank transfers, money wires, mailing physical cash, meeting in person, etc, etc. Typically the buying of crypto is safer because you know the crypto you want is sitting in a third-party escrow wallet and as long as you can prove that you delivered the money, you get your crypto. If there is a dispute a neutral third party will review your evidence and send the crypto to whoever deserves it. There are tons of websites that offer this like localbitcoins.com or localetherium.com with varying levels of anonymity. Also I recommend downloading Bisq which is a program that facilitates this exchange in the same way and is by nature very anonymous.