It is possible to sign in to Day One Sync with Apple ID. This is to help users sign in without having to remember yet another login.
You will not be able to add an email address to your account with these steps if that email is already associated with a Day One account.
Click Sign In with Apple ID.
Log in with your Apple ID. If the Day One account uses the old way of singing in with Apple ID, you may need to authenticate a second time after you see an error.
Here you can add an email address and a password.
You will receive an email to confirm this. Click the link to confirm and sign in with the option to Sign In with Apple ID again.
Now you can use that email and password to sign in with you choose "Day One account" to sign in on the apps.