Have you even gotten under the car to see if there is something rubbing, like your exhaust or heat shield in that area?
That's what it sounds like to me.
Your shifter won't make that kind of sound and if it was loose enough to rattle that bad then you'd feel it while shifting.
Get under the car and see what you're hearing.
It might be something as simple as a loose heat shield for the exhaust in that area.
Since these shifter modules (well, that's what it is, a module) don't sit directly on the transmission, they are linked by cables. I'm wondering if for some reason the clips holding yours have have gotten worn over time. It sounds like a transferred vibration. Or it could be something to the extent of what misnblu said.
Is the sound, with clutch engaged? Or disengaged? Maybe clutch plate, pressure plate, release bearing? Remembering when you start it you press the clutch (disengage). So the sound starts after you foot off the clutch (engage). This may help you eliminate a clutch issue.