It's not paid by taxes but by health insurance. We have both a private and a semi-public system (managed by "Krankenkassen", independent insurance entities, that compete for members; it's... complicated, but less so than in the US, and it does not depend on the employer).
Some might say it's like a tax, it gets deducted from your paycheck, but then that's the same everywhere, including the US. Difference is that it's managed by independent entities and not the government, and they even compete. The private insurances are quite significant in Germany too. You are allowed to leave the government-mandated "Krankenkassen" if you earn over a certain amount of money per year, then you have to get private insurance - which is less expensive if you are young, but unlike the Krankenkasse family members (e.g. non.working spouse, children) are not included and it may become much more expensive when you get older, it's hard to predict because it depends on many factors including if you picked the right insurance company (that didn't lose as much money and now has to raise prices).
I don't know if I would understand the German "insurance" system from that explanation. Maybe to complement your explanation another try at it:
Germany has two insurance-like systems to pay for medical treatments:
(1) insurances, with fees aligned with damage risk associated with the policy holder.
(2) a system of "funds" that provide insurance benefits for insurants, however, are not organized like an insurance.
The system (2) is quite old (dating back to the 19th century) and has a few peculiarities:
* fees vary by income of the member of the fund (principle of solidarity, basically people with well paying jobs subsidise others).
* half of the fee for a member is payed by the employer, half by the employee, deducted from the paycheck.
* family members can be insured, too, without increasing fees.
* fees are not dependent on your risk, i.e. they do not increase with age, preexisting conditions, etc.
The alternative to this fund are insurances that work on a risk-based pricing scheme - item (1) in the list:
* in order to be able to sign up for them, you need to have a certain steady income above a threshold (above 50000 EUR I believe).
* pricing is per insurant, i.e. extra insurance contributions for kids and family members if they are on the same plan, etc.
* for young adults these plans are usually cheaper than for older people, which means you can get "trapped" in a contract that is more expensive than the public funds I wrote about above.
For both systems, treatment prices are kind of fixed (as a rule of thumb, doctors can charge the insurance a bit more than the public funds, by a constant factor of 2-3, but I heard from doctors that this depends on the treatment).
Some might say it's like a tax, it gets deducted from your paycheck, but then that's the same everywhere, including the US. Difference is that it's managed by independent entities and not the government, and they even compete. The private insurances are quite significant in Germany too. You are allowed to leave the government-mandated "Krankenkassen" if you earn over a certain amount of money per year, then you have to get private insurance - which is less expensive if you are young, but unlike the Krankenkasse family members (e.g. non.working spouse, children) are not included and it may become much more expensive when you get older, it's hard to predict because it depends on many factors including if you picked the right insurance company (that didn't lose as much money and now has to raise prices).