When you setup a dedicated monitoring cluster for your existing production cluster, make sure that the elected master node is the first to send metrics (via metricbeat) to the monitoring cluster. (It is totally fine if all the nodes are publishing metrics in the same time, just make sure that the elected master is among them). As it turns out, the e.m. publishes not just metrics, but information needed for the monitoring cluster to setup Stack Monitoring UI.
The step-by-step documentation is pretty good, it just fails to mention this tiny detail which can result in hours of debugging. Ask me why I know 🙄