Q:

Where is the Registry running

I can create a container running a registry: docker run -d -p 5000:5000 --restart=always --name registry registry:2
But docker has a default registry, I can see that the registry is at Registry: https://index.docker.io/v1/ and it must be local, but where is it - do you know?
It is correct that if using a browser and go to: https://index.docker.io/v1/ it will take you to docker hub: https://index.docker.io/v1/
But all my local images is local on my machine, so there must be some where the registry is running.
You can see the registry if you do:
docker system info
Containers: 32
 Running: 29
 Paused: 0
 Stopped: 3
Images: 205
Server Version: 18.06.0-ce
Storage Driver: overlay2
 Backing Filesystem: extfs
 Supports d_type: true
 Native Overlay Diff: true
Logging Driver: json-file
Cgroup Driver: cgroupfs
Plugins:
 Volume: local
 Network: bridge host ipvlan macvlan null overlay
 Log: awslogs fluentd gcplogs gelf journald json-file logentries splunk syslog
Swarm: inactive
Runtimes: runc
Default Runtime: runc
Init Binary: docker-init
containerd version: d64c661f1d51c48782c9cec8fda7604785f93587
runc version: 69663f0bd4b60df09991c08812a60108003fa340
init version: fec3683
Security Options:
 seccomp
  Profile: default
Kernel Version: 4.9.93-linuxkit-aufs
Operating System: Docker for Mac
OSType: linux
Architecture: x86_64
CPUs: 2
Total Memory: 2.934GiB
Name: linuxkit-025000000001
ID: Q6IO:V5CP:OHJL:4KJP:ZG2X:GV5W:YHMM:2WCK:4V4O:O6T3:A4E4:BJHM
Docker Root Dir: /var/lib/docker
Debug Mode (client): false
Debug Mode (server): true
 File Descriptors: 206
 Goroutines: 223
 System Time: 2018-08-29T11:56:34.8224409Z
 EventsListeners: 2
HTTP Proxy: gateway.docker.internal:3128
HTTPS Proxy: gateway.docker.internal:3129
Registry: https://index.docker.io/v1/
Labels:
Experimental: true
Insecure Registries:
 127.0.0.0/8
Live Restore Enabled: false

A:

That is the default registry which is dockerhub:
https://hub.docker.com/
Also see:
https://github.com/moby/moby/issues/7203
You cannot change the default registry (which is dockerhub). What you can do is push and pull using your registry as a prefix.
For example:
docker push localhost:5000/yourimage

docker pull localhost:5000/yourimage

As per my comment below - this registry runs locally and with
docker ps | grep registry:2

you can see it running. You can then use it's id to get the logs where you will see the activity. 
You can also make use of the api by doing a call to:
curl -X GET http://localhost:5000/v2/_catalog

This will list all the images you have pushed to your local registry.

