Do you always have to use Cognito when implementing Web Identity Federation?

Hi, no it is not mandatory to use Cognito, just recommended by AWS! 

There are alternatives such as Auth0 which could  be used instead – in fact at ACG we actually used Auth0, only because when our website was first built, Cognito was not as feature rich as it is now.  

https://docs.aws.amazon.com/IAM/latest/UserGuide/id_roles_providers_oidc.html