I honestly wouldn't use such a tool for anything mid to large size. Git is kind of a nightmare. Now, an SQL database, or a distributed key-value store, I would use as a backend.
I don't believe they mean they're using git as the data store for the application. Sync with git means updating tickets based on parsing action commands embedded in commit messages pushed to your git server. Jira can actually do this, too, though I think you need to be using Bitbucket as your git server.
Here are the issues though with Jira's Github integration:
1- Requires unito or an additional plugin to truly function
2- Needs quite a bit of setup to get minimal value
3- No built-in dashboards optimized for git events (ie when is a PR merged, predicted vs actual, how does that relate to your team's current sprint).
4- One-directional sync (vs a true bi-directional sync).
We've optimized for built-in dashboards, minimal setup and a bi-directional sync. Over time, we're also going to be building inside the terminal or inside Github's interface, to truly enable seamless interactions with Git.