How can I get the identity id of the user (logged in by AWS Cognito) that invoked an AWS Lambda function? Do I have to use the SDK on the Lambda function to get the identity id?
Per the docs, it looks like information about the identity provider would only be available for an invoke through the Mobile SDK.
To get around this, one option is to pass the identity ID to the function manually as part of the event. Assuming you are doing something like AWS.config.credentials = new AWS.CognitoIdentityCredentials(...) then you should be able to get the ID via AWS.config.credentials.identityId (after the credentials are refreshed).
EDIT: A better option for identity validation is to let Cognito/IAM handle it, and assume that if a user can successfully invoke a Lambda function, that means they are allowed to. In this case to manage per-user validation, take a look at whitelisting.