Imagine if it were a real monopoly, if there was only one insurance company. And you couldn't even refuse to pay your "premiums", because it had merged with the government, and the premiums were your taxes. Wouldn't that be a nightmare scenario?
A monopoly does not imply there is just one insurance company.
What it means is that you are unable to negotiate fairly because there is a monolithic control working against you.
That can be price fixing by many companies, or a government mandate.
It just means something is monopolizing your choices.
For example, a mafia protection racket is a monopoly by the threat of force if you do not comply.
But the government is not a monopoly because you get to vote and the people collectively control it.
It works for you instead of trying to gain profit out of you.
And public health care in no country has at all precluded private insurance or health care.
It just provides a minimal safety net as competition, in order to break the health insurance monopoly.