No Jason, we may have misinterpreted it.
Normally you have a community / dating website and your software will be integrated and it will use the database of the website itself.
What we thought of is that the community / dating website software is not installed, but purely your software on the domain itself.
So that it is actually a chat website where everyone can log in and chat with others
We have this kind of solution, where our site is based on a webchat. It's completely run with Arrowchat.
So yes, it is possible to make just a chat site with no integration to any existing platforms. But if there's gonna be other than just guest users, then there should exist at least simple username creation procedure.
We have done exactly this - we have a simple registration form, wher user can register an username and password. Then they can use the chat by logging in. That kind of registration form is very easy to do with PHP. Later we have added more functions, like user profiles etc, but that's not necessary.