For the JS logging you see in some of the screenshots, we use some JS on the client side that watches for errors and fires off a request to us with as much data as on the error as we can collect (in most cases we can get stack traces, which is pretty nice).
In terms of where this all gets stored and aggregated, all of our application level logs are shipped into Elasticsearch, and that's where the aggregations and counts you see in the screenshots come from. All of the application level logs are sent in with a unique identifier per-client so that we can keep track of them that way, as well as tagged by username, IP address, and anything else that might be useful in cross-correlating with logs from the rest of our stack.
If there's any more specific details you're curious about, I'd be happy to elaborate.