What do you mean by anonymous connections? Without them being logged in and you actively tracking them, it is anonymous. You'd be reinventing the wheel to de-anonymize the user if you want to track users across devices, which is certainly not anonymous: existing companies use advertising IDs or cookies, based on the problem. There is no way you can identify users across devices devices (or solve this problem better) than Google and Facebook, since you run in 1 application, they run in almost all of them.
"We did look at syncing all this with RTC data channels,", that's when you use a reliable service with additional functionality like history and presence, not WebRTC data channels, that might be why you struggled. It sounds like you should be using WebSockets for this type of data.
It sounds like you're trying to build chat for ecommerce websites, but isn't that Intercom, tidio.com (free tier alternative). Agora is lower level, but also solves these problems and more: messaging, audio, video calls. I don't think any of these offer cross device identification without having users log in on all their devices.