I thought that was interesting, so I just checked - the EU as a whole has about 52.5 million people living in countries they were not born in (including other EU countries) and a total population of about 512 million. So that's about 10%. The US has about 44.5 million foreign born in it, with a total population of about 328 million. So that's about 13.5%.
So no, the EU does not have more immigrants than the US per capita. If you remove European citizens moving to other European countries, they don't have as many immigrants by sheer number, either.
There is also a sliding scale in EU. Western European countries like Germany, France, Sweden, Netherlands, take in far more immigrants than Poland, Hungary etc. The point that triggered this whole thread was immigration is the reason America doesn't have the same quality of life as EU. Which is clearly not true, considering Germany and NL takes in very high % of immigrants. Even with your 13 vs 10% is negligible for any meaningful policy differences. US is different not because of immigration, but due to its political bent.
So no, the EU does not have more immigrants than the US per capita. If you remove European citizens moving to other European countries, they don't have as many immigrants by sheer number, either.