I've been having this issue for a while on my 1999 Volvo V40 T4 manual. Whenever I shift and release the clutch quickly, the car often does a nasty bounce. Let me try to describe it with much detail as possible:
Let's say I start going calmly in 1st, rev up to ~6000rpm, push in the clutch and switch to 2nd. Now if I release the clutch quickly, instead of shooting straight forward the car hesitates a bit for a split second, then shoots forward, then hesitates again less and shoots forward again. All of this happens in one fluid "bouncing motion", as if the car was made of jello (the engine sound slightly accompanies it as well, as if a little bit of load is removed during the bounce).
Sometimes it's softer, sometimes harder, I think it depends on how well the revs are matched. Problem is much more prominent in low gears, especially first to second. To avoid this shifts from lower gears must be done extra gently, to allow more clutch slippage.
The car feels a bit "loose" in general when driving. For example even if you take off from a dead stop, the start is never smooth - you can clearly feel the moment when the clutch stops slipping. Feels like as if at that exact moment torque is lost and then appears again some milliseconds later, also a sort of "bounce". If I take off my foot off the accelerator while in a low gear and then step on it again, the car jerks rather aggressively, resembling the bounce a bit.
The bouncing feels fully mechanical, though. I don't think it's the engine misfiring, but rather some kind of looseness in the drivetrain. I always thought the reason for this was torn upper engine mounts (the car has 4: two upper, fixing the top sides to the chassis and two lower, fixing the center to the crossmember), however replacing them made things better only slightly, so I'm quite stumped here. The lower mounts are looking good. If one puts the car in gear and pushes it back and forth a little with the hood open some clearly noticeable engine movement can be seen, as if the top mounts are a too soft. Not sure if this is the cause, could make a video if that might be the culprit.
This model has a dual mass flywheel, but it emits no classic rattling noise on idle that indicates spring defects. Could a worn DMF cause this? Any ideas what else it could be are truly appreciated, don't hesitate even if you're not sure, I'm running out of ideas here.
EDIT: got a second pair of eyes to check the suspension for me. Everything seems in order as expected, was suggested that this may be caused by a worn DMF, gearbox or differential.