Apple Id locked and forgotten birthdate

As said above Eventually you will be able to enter account recovery if you want.

See Recover your Apple ID when you can’t reset your password https://support.apple.com/en-us/HT204921. Read the whole document carefully, not just the excerpt below.

“Account recovery might take a few days or longer, depending on the specific account information you can provide to verify your identity. After you request account recovery, you'll get an email with a confirmation of your request and the date and time of when you can expect to regain access. … When the wait period is over, Apple sends you a text or automated phone call with instructions to regain access to your account.

You need to read what I mentioned above. You need to enter account recovery if you want your account back

See If your Apple ID is locked or disabled https://support.apple.com/en-us/HT204106 "If you or someone else enters your password, security questions, or other account information incorrectly too many times, your Apple ID automatically locks to protect your security and you can't sign in to any Apple services.

If you use two-factor authentication, you need a trusted device or trusted phone number to reset your Apple ID password and unlock your account. If you have a recovery key, you can also use it to help you regain access to your account. …

If you use security questions with your Apple ID, or if you don't have access to a trusted device or phone number, go to iforgot.apple.com. Then you can unlock your account with your existing password or reset your password. After multiple unsuccessful attempts to unlock your account, your Apple ID will remain locked and you can try again the next day.” Eventually you will be able to enter account recovery if you want.

See Recover your Apple ID when you can’t reset your password https://support.apple.com/en-us/HT204921. Read the whole document carefully, not just the excerpt below.

“Account recovery might take a few days or longer, depending on the specific account information you can provide to verify your identity. After you request account recovery, you'll get an email with a confirmation of your request and the date and time of when you can expect to regain access. … When the wait period is over, Apple sends you a text or automated phone call with instructions to regain access to your account.