Unable to push docker image into GCP container registry [permission error]
It is likely that your account doesn't have permission to store/write the docker image in GCR. The GCR in GCP is backed by a Google cloud storage bucket and your account needs a write permission on the bucket.
For eu.gcr.io, the equivalent bucket is gs://eu.artifacts.$PROJECT_ID.appspot.com.
Try to grant your account, a roles/storage.objectAdmin role at the bucket level. Of course, you can do this at the project or org level as well.
You can find details - https://cloud.google.com/container-registry/docs/access-control#permissions