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 add-on 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.