Using Visualforce Page, we can get the First Name and Last Name of the user using {!$User.FirstName} and {!$User.LastName} expressions. sessionStorage can be used so that we can access the value from the Experience Cloud Site "Snippet Settings File" JavaScript. We can pass the Chat Visitor from the Experience Cloud Site "Snippet Settings File" JavaScript. In the Einstein BOT, we can use Context Variable to get the Chat Visitor name and greet.
Blog Post:
https://www.infallibletechie.com/2023/09/greet-customers-in-salesforce-einstein-bot-without-pre-chat-from-experience-cloud-site.html