Determining who has the highest standard of living even today is difficult given how hard it is to define "quality of life." This problem is made much worse when we try to look at standards of living over 200 years ago. In those days, there were very few records kept, which makes it very difficult to make any objective comparisons.
Those who say that the American standard of living was highest generally point to things like the physical size of American people. Americans were (it is believed from things like military records) quite a bit larger on average than Europeans were. This implies that Americans had more and better food and were generally healthier than their European counterparts. If we define "standard of living" in this way, then Americans had the highest standard in the "European" (Europe and other places peopled by Europeans) world, at least.