Q:

Is Azure Resource Manager equivalent to what kubernetes is for Docker

Can you think of Azure Resource Manager as the equivalent to what kubernetes is for Docker?

A:

I think that the two are slightly different (caveat:  I have only cursory knowledge of Resource Manager)
Azure Resource Manager lets you think about a collection of separate resources as a single composite application.  Much like Google's Deployment Manager.  It makes it easier to create repeatable deployments, and make sense of a big collection of heterogeneous resources as belonging to a single app.
Kubernetes is on the other hand turns a collection of virtual machines into a new resource type (a cluster).  It goes beyond  configuration and deployment of resources and acts as a runtime environment for distributed apps.  So it has an API that can be used during runtime to deploy and wire in your containers, dynamically scale up/scale down your cluster, and it will make sure that your intent is being met (if you ask for three running containers of a certain type, it will make sure that there are always three healthy containers of that type running).

