If the user signup on a website, will he have to login again in the chatbot (if it requires authentication) and vice versa?

After logging in to our website, is it required for the user to log in again into the chatbot while initiating a conversation?

No, the users need not have to login again.
In the webSDK, developer could configure the index.html to pass to customData along with botinfo which will be available under BotUserSession. If the services used at chatbot, need any information of the user to hit the endpoint, developer could configure the service request to take user information from customData passed from webSDK.