AWS Cognito username/email login is case-sensitive

You could trigger a Lambda function after sign-up to change the email to lowercase. Without actually testing it, you should be able to trigger a Lambda post confirmation. That Lambda could use AdminUpdateUserAttributes API, called from your SDK of choice, to change the email to lowercase.

Note that user names are also case sensitive.

Fixed on new user pools. You can turn off case sensitivity now.

https://aws.amazon.com/about-aws/whats-new/2020/02/amazon-cognito-user-pools-service-now-supports-case-insensitivity-for-user-aliases/