Does the American public know, that their perception of left and right
is is skewed and right-shifted in comparison to many other Western
The American public is mostly oblivious to the domestic politics of countries other than their own, although there is some vague familiarity with the leading political parties of Canada, Mexico and the U.K. These topics receive very little U.S. media coverage and are rarely discussed by U.S. politicians.
Politically active, college educated, progressives (perhaps 10% of voters, often via support for Bernie Sanders rather than knowledge of European politics directly) tend to be aware of the array of political parties on the left in Europe and to have a favorable opinion of Democratic Socialism and the Green Parties. A narrower swath of educated politically active people on the far right and for whatever reason, quite a few libertarians (perhaps 5% of voters in all), have some meaningful awareness of the parties of the far right in Europe, but less awareness of European Center-Right parties.
Or do they assume that European conservative parties refer to policies
similar to the Republican parties policies?
I doubt that 5% of Americans have ever had any thoughts on this subject, and would have a hard time even associating famous European national leaders with particular political parties. The very internationally aware without first hand experience mostly might have some dim sense that there might be differences.
Maybe 1-3% of Americans might have some generally familiarity with European political parties and what they stand for. Those who do would mostly have either studied it academically in college, or served in the military in Europe, or otherwise lived in Europe (e.g. as an exchange student or expatriate employee).
I'm starting with a rough sense from surveys (also here and here) and exit polls of the breakdown of Americans by politics and education, I'm cross referencing that (intuitively) with categories of people I've experienced knowing about or discussing these things and prevalence of knowledge within categories, after lots of experience in politics at the grass roots level and in media quotations of people and politicians, I'm factoring in (intuitively) data on levels of depth of knowledge to similar degrees, about Europe or foreign affairs generally, It's an informed guesstimate, but order of magnitude close.