What consequences could occur if I made a Microsoft account with fake information for privacy reasons?

The main thing that really matters for account and game usage is your email address. I highly doubt Microsoft will do anything to you for putting in fake info, as it's not their problem if you do... it's yours (unless you are stealing someone else's identity, in which case they will act if the real person you are stealing from notices and opens a case).

You will probably face issues with account security and recovery, as setting up things like two-factor authentication requires your actual phone. If your account is hacked, you can't prove it is yours since you will need to provide proof you are the account owner (since this information is faked, any proof you provide will only prove you aren't the "real" owner).