More experiments have been done. Firstly, the correlation with chromecast_1, TV_1, and the GoogleChrome-Kaspersky warnings have been repeatedly confirmed using two different computers. I removed chromecast_1 (from TV_1) and wondered whether my PC could communicate via the micro USB port used to power the chromecast. Apparently not. However, while the chromecast was powered by my PC, my wife called out that the KIS warnings had returned to PC_2. The warnings stopped being generated when the chromecast was unplugged from PC_1.
Next I wondered why chromecast_2 was not involved. It had been powered up behind TV_2 all this time not causing any problem. I connected chromecast_2 to TV_1 and the Kaspersky warnings returned to PC_1 & PC_2 when opening GoogleChrome. The issue had more to do with TV_1 than which chromecast device was connected.
Here are some missing clues from the above description. TV_1 has an ethernet cable to the Fritz!Box 7490, TV_2 does not. PC_1 also has an ethernet cable connection to the Fritz!Box, (PC_2 uses the wifi). The problem vanished when I removed the ethernet cable from TV_1. TV_1 is not very "smart", so we use the chromecast exclusively these days. Removing the ethernet cable from TV_1 incurs no loss of functionality, so this is the solution for us. However, the mystery remains.
Further speculation. Why did this issue suddenly appear around 16 hours ago? There is one possibly related event. I was trying to get Windows 10 file sharing to work properly over the network. As part of getting a clean start for the network (and removing leftovers from the old Homegroup), I rebooted the Fritz!Box 7490 VDSL modem. Although I had powered down the PCs, the 2 chromecast devices and 2 printers were temporarily without a network connection via the wifi or cable. It is likely, the Kaspersky warnings about self-signed certificates began after this event. Can we make it go away again? In the meantime, disconnecting TV_1 from the ethernet seems to be working.
If somebody can make sense of this phenomenon, I will be most grateful and very impressed.