If the revision checked out has a git tag associated with it, the tag name will be exported during the build as GIT_TAG_NAME.
If a message was specified when creating the tag, then that message will be exported during the build as the GIT_TAG_MESSAGE environment variable.
If no tag message was specified, the commit message will be used.

If the revision has more than one tag associated with it, only the most recent tag will be taken into account, unless the refspec contains "refs/tags/" — i.e. builds are only triggered when certain tag names or patterns are matched — in which case the exact tag name that triggered the build will be used, even if it's not the most recent tag for this commit.
For this reason, if you're not using a tag-specific refspec but you are using the "Create a tag for every build" behaviour, you should make sure that the build-tagging behaviour is configured to run after this "export git tag message" behaviour.

Tag and commit messages which span multiple lines are no problem, though only the first 10000 lines of a tag's message will be exported.