The Solidaritätsprinzip is named that to distinguish it from the Äquivalenzprinzip (equivalency principle) of private insurance: with private insurance, your cost rate is determined by the services you would like to have covered as well as the assessed risk that you will consume these services as calculated by the insurance carrier's actuaries. In Germany (and other European countries), the public option insurance instead only uses your income to determine your rate: the insured risks are common and paid by everyone together instead of by the individual.

"Solidarity" is a pretty good term for that abstract idea. What would you like to suggest instead?

And yeah, I understand that people that do not buy into the concept of a society pooling together such risk to create a model that works best in the aggregate instead of emphasizing individual choice are going to hate everything about this. If you want to be able to save money in exchange for taking on risk, such a society can't let you easily do that because it'll inevitably lead to a tragedy of the commons; such a society would also likely help you anyway if you made the wrong risk trade-offs and become a burden to the state.

Also, let's be clear: this principle is still very popular in Germany. I haven't lived there in a long time so I was curious and gave it 30 seconds. Admittedly the source here is an insurance carrier, but they found in a survey that while 66% of Germans last year were worried about insurance costs and the financial sustainability of the German healthcare system, 73% support the solidarity principle (https://www.aok.de/pp/gg/update/solidaritaet-gkv/)

Why not call it taxes?

And there certainly is consent, given voters can choose to elect people who will change the system, or not.