Do you think the shake you can see in the video is normal or do I have some issues with the gimbal?
I also had stronger shakes below 20m of altitude without perceivable wind, so I don't think it could be related to strong wind.

Hi there. Sorry about the trouble. If the drone crashes or experiences any collisions, we recommend sending the drone for diagnosis. If not, can you please follow the steps below? (feel free to skip some steps if was already done:
1. Make sure to remove the gimbal protector before powering on the aircraft to prevent the gimbal motor from being stuck, or the gimbal will become shaky.
2. Please make sure that the gimbal is in Follow mode. If it is set to FPV mode, the gimbal view will be tilted when the aircraft tilts.
3. Please check whether the aircraft is in Sports mode (S-mode) when the gimbal becomes shaky. In S-mode, greater emphasis is placed on flight performance. During flight, when the aircraft attitude is too large, the gimbal attitude stabilization cannot be guaranteed. If the gimbal is shaky in S-mode only, which is normal, please switch the flight mode of the aircraft to Normal mode (N-mode) or Cine mode (C-mode) to achieve a more stable view.
4. Please check whether the wind was strong in the flight environment. If the wind is too strong, the aircraft will vibrate, resulting in the image transmission screen being shaky. We recommend that you test the gimbal in a windless or breezy environment.
5. Please perform gimbal calibration in the app. Using with DJI Fly: [Settings -> Control -> Gimbal Calibration].
If the steps above were done and the issue persists, we advise submitting an Online Repair Request and sending the aircraft back to DJI for further diagnosis. Have a great day ahead.

Ofcourse it's not related with the wind because the gimbal will try to smooth out the movement even if there is a very strong wind and the drone is tilted more than the limitations of the gimbal. But if you are flying the drone in a very cold environment (less than 10 degree celcius) the gimbal may operate a bit worse. In a very cold environment the gimbal may do strange things.

I'd say gimbal problem, but it would be good to see more than 2 seconds : o )

Not much to go on with that video but i don't think its the props or motors.

At a guess, its taken a knock (crashed) and something is either broken or come loose with regards to the gimbal.

my dji mini 2 se is smooth from start to finish so would say this is not normal. Have you tried recalibrating the gimble ?

At the end I asked for a replacement under warranty and the new drone doesn't have any issue. With the previous one I tried to calibrate everything and the issue still persisted. Of course I never had any crash with it and I only had a couple of flights.