I really don't mean to be rude or anything but I think you're missing the point a little bit. The site you linked to is just a discord widget that embeds a server onto the page. It looks like a user can't message anyone without logging in with discord. And when they do, they'll just be participating in the convo happening in the server.
Chatlike is a support widget. Any user can message a business from the widget, embedded on their site, and they can respond. Like Intercom, Drift, Crisp, etc.
Chatlike is actually somewhat inspired by Chatlio. It does the same thing, but with Slack. Check their prices :)
We do over at ChargeOver. Chat is often a lot quicker than phone calls or emails.
We've had great experiences with Chatlio - https://chatlio.com/ It integrates into Slack, so we can chat with customers directly from within Slack (each new chat spawns a new Slack channel automagically).
We originally tried Intercom.io, but at the time they didn't have a Slack integration. Slack just makes it SO much easier since we're in there already - one less thing to monitor and keep track of.