You may receive the following error while installing the 1.0.5 firmware update for the Dell TB16 docking station:
Dock NVM - Thunderbolt Dock Firmware Update failed:

When you try to run the following command manually you may get an error that the firmware update tool did not find a (suitable) Thunderbolt driver: 

• FWUpdateCmd.exe UpBme

The firmware update tool requires the older Intel Thunderbolt Controller driver version 17.4.79.510, A12 for Windows 10 version 1909 (19H2) or lower, dated March 6, 2019:

• Intel-Thunderbolt-Controller-Driver_TBT79_WIN_17.4.79.510_A12.EXE
  

So first you need to uninstall the newer version, e.g. 1.41.1193.0, A11 dated November 5, 2021:

Next reboot and install the older Intel Thunderbolt Controller driver version 17.4.79.510, A12.  The first installation attempt(s) may fail.  Reboot again and try a couple of more times until driver is successfully installed.

As the driver is only supported on Windows 10 version 1909 (19H2) or lower, you may need to use another Dell laptop with an older Windows 10 version and an USB-C/Thunderbolt port to be able to update the firmware.

Once the older Intel Thunderbolt Controller driver version 17.4.79.510, A12 is installed you can try and install the 1.0.5 firmware update for the Dell TB16 docking station:

If you are running a Windows 10 version newer than 1909 (19H2), then you can uninstall the older Intel Thunderbolt Controller driver version 17.4.79.510, A12, and install the latest version 1.41.1193.0, A11 dated November 5, 2021 instead:

• Older version: Intel-Thunderbolt-Controller-Driver_TBT79_WIN_17.4.79.510_A12.EXE
• Latest version: Intel-Thunderbolt-Controller-Driver_TBTC0_WIN_1.41.1193.0_A11.EXE

Conclusion: you may need to replace your Thunderbolt Controller driver version with an older/specific version to be able to update the TB16 docking station firmware.