I had the same issue, during idle on neutral still would get vibration feeling while holding the gear shifter, I used to own the same car (2013 A4 b8.5 manual) but at the time I wasn't sure where it was coming from.
It was annoying to me at times so i took the car to the dealership but i told them i felt the vibration through steering, as its electronic steering now, they thought i was crazy, the end.
currently I own 2016 A4 B8.5 Manual and a whole lot of crap that used to annoy at me on the 2013 model seemed fixed now. To name another, the rpms spiking a bit too high while switching from 1st to 2nd thus creating a jerky feeling. I purchased eurocode tranny mount inserts and that actually helped a lot on the jerky feeling issue. To think of it, it wouldnt be an overreaching assumption to say the inserts actually did help with the vibration feeling as well.
I re-installed the Eurocode tranny inserts on my 2016 B8.5 model, of course. And I wasnt blown away from the mod as the car felt already pretty fixed.
On the vibration subject, I drove a bmw 328I manual with the 2.0 turbo and it had an even worse vibration while idling.
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