For better or for worse, manager's productivity is normally taken from the developers/engineers the manager manages, namely the team's overall productivity, so we're back to the same problem of how to measure developer's productivity
My productivity (as a manager) is measured by my ability to deliver on commitments I make with product management and senior leadership. Basically, my ability to match my teams' productivity with corporate goals.
Which lines up with nradov's comment about the smallest useful unit for measuring productivity is the typical self-contained development team. Attempting to determine if Bob in TeamA is more productive than Cindy on TeamB doesn't generally result in any actionable information. What matters (from a senior leadership perspective) is can TeamA or TeamB build the things that need to be built in a timeframe that's acceptable to stakeholders.
As a manager, if I feel like Bob or Cindy are unproductive, then I need to figure out why. And LoC or number of commits isn't going to tell me that. Possibly the number of defects found in QA, but even that isn't perfect.