Why can't I withdraw my earnings to any wallet of my choice?
Brave's business model is considered to be revenue sharing, and the AML (anti-money laundering) laws of the US and some other countries require recipients of shared revenue to be identified by KYC.
Users who wish to use their BAT within the ecosystem to tip etc. do not need to perform KYC. Only users who wish to use their earned BAT as real cryptocurrency or convert their BAT to fiat currency must go through the KYC process.
Right now the only way to withdraw your Brave Rewards from the browser is through BitFlyer for Japanese users, or Uphold\ and Gemini for other users. Please check each exchange for the current state of support in your country (here for Uphold, here for Gemini).
Eventually the Brave Wallet is planned to integrate a KYC process, which will allow users to transfer their Rewards to any Ethereum address of their choice. The release date for this feature being added to Brave Wallet is unknown. [Current February 23 2022]