Is your foot on the brake or off?

The sound is too mechanical to be coolant or any other fluid.

Can you feel anything through the gear lever?

Things that come to mind are the clutch throw our bearing.

Check the rpm at which the noise occurs. It sounds very much like a resonance thing. Something is spinning up and as it goes through this frequency it is vibrating internally or hitting something externally.

I had something similar with my Hiace. When I started it up for the first time in the morning, as it went through around 1000 rpm, there would be a horrible buzz that would go through the van. It turned out to be an exhaust clamp that had vibrated around just a few degrees to a point where with the exhaust cold, one of the clamp bolts would just make contact with the chassis rail. No point checking this when warm as the exhaust had expanded enough to move it clear of the chassis.

The solution was easy in my case.

Chances are that it is something like this (not necessarily the exhaust) in yours.