AWS Cognito identityId and identityPoolId is null

I have been trying Cognito user pools with federated identities and identity pool. I created identity pool and assigned cognito as auth provider for it. Cognito has Google identity provider. I used Amplify to sign-up/sign-in, it's configured with Identity pool and my User pool. When I log in, identity is properly created in identity pool and linked to user in user pool. I can see identityId and identityPoolId in credentials available in Amplify session, both contain valid information, I was able to invoke Cognito sync and exchange identity data with no problem.

I have problem with identity sent to API Lambda handlers integrated via API Gateway. I use REST API (old one), integrate as Lambda Proxy and I created Lambda authorizer, which simply dumps event into log.

When I check CloudWatch logs, I see

"identity": {
            "cognitoIdentityPoolId": null,
            "accountId": null,
            "cognitoIdentityId": null,
            "caller": null,
            "sourceIp": "185.5.69.158",
            "principalOrgId": null,
            "accessKey": null,
            "cognitoAuthenticationType": null,
            "cognitoAuthenticationProvider": null,
            "userArn": null,
            "userAgent": "Mozilla/5.0 (Macintosh; Intel Mac OS X 10_15_7) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/103.0.0.0 Safari/537.36",
            "user": null
        },

Why us cognitoIdentityPoolId and cognitoIdentityId null? I use Amplify API integration and I see authorization header with pre-signed request in logs.

What is required to get access to identity in lambda authorizer?