Red cabbage or Blue Cabagge, in northern Germany is also called red cabbage (Rotkohl), same color for some regions of Austria and Switzerland (Rotkraut). In southern Germany it is called Blue Cabagge (Blaukraut). If you ask people in Germany what the color of the vegetable is, they will answer "purple" (lila). There are some strange ways in the evolution of a language, depending on the region and events in the region…
This particular cabbage is coloured by a chemical that responds to the pH of the stuff it comes into contact with. The colour can range from quite bright red to quite clear blue, and even green or yellow.
It's perfectly possible for the red cabbages to be turned into blue dishes, and the pH of the soil will also have a large effect on the colour produced by the plant. You can see on various stock photos how the plant has a clear blue hue (before harvesting, at least): https://www.alamy.com/stock-photo/rotkraut.html?pseudoid=562... Even ignoring the leaves, the outside of the parts that generally get cooked have a clear blue hue in many pictures I can see.
The reason "lila" wasn't used to describe the cabbage is that the German language lacked a word for it. It entered the German vocabulary somewhere in the 19th century (https://en.wiktionary.org/wiki/lila#German). In a similar fashion, there was no separate word for orange in many European languages until somewhere around the 15th-16th century, when oranges (the fruit) were starting to get imported. You had yellow-red and other hue combinations, but it wasn't as separate as it is today.
I'm sure there were people who used "Rotblaukohl", but it makes sense that only one colour remained.
As for the carrot, orange carrots were actually not all the common for centuries. The original plants now known as carrots were imported to Europe from the middle east and cultivated in the Netherlands, but orange carrots weren't all that common in Europe before the 16th/17th century. The base plant of the orange carrot was actually white/yellow and got its orange outside hue quite some time later, after selective breeding. I wouldn't be surprised if the carrots that were first exported to modern German areas were still yellow in colour. Gelbrübe for orange carrots makes a lot of sense, historically.
I do like the "purple yellow root" name, especially since the first carrot cultivars to reach Europe (long before orange/yellow carrots) were actually purple. I don't think they received quite the popularity carrots received, at least not much further north than the Mediterranean.