Is it true?

It is true that in the U.S. one only pays Social Security tax up to that $97K figure.
After that they do not pay Social Security Tax.

I thought that statement was very clear.

If the opposite was true, no one would be paying Social Security tax on their first $97K of earnings.

Exactly so. Medicare tax is paid on all earned income.

I see. So it means once a millionaire guy has paid up to that amount in tax, he has fulfilled his obligation, and he will no longer pay taxes however much he may earn later.

The 2010 rate for social security is limited to $$106,800 in income.

You are correct except there are some employees who do not pay in to SS or Medicare. It was challenged in court, at least in WA State, and a Federal court judge ruled in favor of those particular employees.

I think they have to be Amish, or part of some other group that assures their care. Do I have that right?

Not just a millionaire, but anyone whose income exceeds that level.

Also, when you say tax, we are talking only about social security tax, not federal, state or city income tax.

Where I live, Texas, there is only federal tax income tax.

In Wisconsin there is federal and state income tax.

In New York city there is federal, state and city income tax.

And of course we are not talking about sales tax. Everyone pays that.

Personally, I believe removing or amending the level one pays of social security tax would be a desireable thing.