Yes, if you navigate to the job workspace, you will find that Bamboo does not do a straightforward git clone "under the hood", and the the remote is set to an internal file path.
Fortunately, Bamboo does store the original repository URL as ${bamboo.repository.git.repositoryUrl}, so all you need to do is set a remote pointing back at the original and push to there. This is what I've been using with both basic Git repositories and Stash, creating a tag based on the build number.
git tag -f -a ${bamboo.buildNumber} -m "${bamboo.planName} build number ${bamboo.buildNumber} passed automated acceptance testing." ${bamboo.planRepository.revision}
git remote add central ${bamboo.planRepository.repositoryUrl}
git push central ${bamboo.buildNumber}
git ls-remote --exit-code --tags central ${bamboo.buildNumber}
The final line is simply to cause the task to fail if the newly created tag cannot be read back.
EDIT: Do not be tempted to use the variable ${bamboo.repository.git.repositoryUrl}, as this will not necessarily point to the repo checked out in your job.
Also bear in mind that if you're checking out from multiple sources, ${bamboo.planRepository.repositoryUrl} points to the first repo in your "Source Code Checkout" task. The more specific URLs are referenced via:
${bamboo.planRepository.1.repositoryUrl}
${bamboo.planRepository.2.repositoryUrl}
...
and so on.