Dell Pro support quickly came up with this being a hardware problem, offering to replace the Motherboard of the laptop. But as the external displays worke flawlessly when connected before powering the laptop, I couldn't agree with this explanation.
A web search for VMM2300 returns one hit from superuser, but this one is for Latitude E7440.
The solution however is very similar: Downgrade the Synaptics VMM2320 MST HUB Firmware to a version lower than the (pre-installed) A05 / 2.22.
In the beginning, I installed A01 / 2.15, which already solved the problem for me.
I did then upgrade step by step to A02 / 2.17 and A03 / 2.21, which do also work.
Installing A05 / 2.22 brings the error back.
So my conclusion for now is: Use the A03 / 2.21 version of the firmware. With this one, I can connect and disconnect external monitors, dock and undock the laptop, restart it and it does reliably recognize each change to the display devices.
Note: The Dell Support pages have quite detailed installation instructions. For each of the firmware installations, it is required to shut down the laptop, either dock it or connect an external monitor to the laptop itself (it depends on which firmware you are going to install whether dock or external monitor is required) and then start the installation.
Best Answer
You will need to get a USB -> VGA adapter in order to run three monitors or a more expensive solution. These USB -> VGA Adapters usually run $50. However, they suck for anything other than business use. Even then, it feels like you're "RDPing or VNCing" into that extra monitor. It is rather laggy.