I use artifactory and nexus, but I wonder what artifact manager could be used in GCP? Is there a managed artifact manager or do I have to run one myself in a VM?
If the artifacts in question are docker images then the recommended artifact management solution is the Container Registry, well integrated with other GCP products producing and/or using such images.
AFAIK they don't have a real artifact manager for other kinds of artifacts, they suggest the rather general purpose Cloud Storage for that. You can find an example of such use in the Quickstart for Go guide. Most likely not what you're looking for:
artifacts: objects: location: 'gs://[BUCKET_NAME]/' paths: ['hello']
A new service have been released this year called Artifact Registry