I want to know if there is a subscription fee for the Gitlab container registry?
The container registry in GitLab is available for everyone. Check the tier marker next to the chapter heading in GitLab Container Registry | GitLab saying
all tiers, which means including the free tier.
Depending on the administrative settings for self-managed instances, or GitLab.com SaaS, there can be storage limits applied to the projects and groups you are using. On GitLab.com SaaS, storage quotes are documented here: Storage usage quota | GitLab