Yes — it does support repositories supporting many projects. That said, the associations are made by the developers when they commit code to a specific issue key (which is part of a project). The permission that allows a user to see these commits in a Jira project is whether they have the "View Development Tools" permission for that project (that's a Jira setting).
The only project permissions that the Git Integration for Jira app has are for the Repository Browser (Git dropdown menu). To associate a repository with all Jira projects or only specific projects, see Repository Browser setting in the Connect to Git Repository wizard or edit a repository in the Git Repositories configuration page.
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