I've been mistaken for Polish in short interactions (believe me, it doesn't last long). The longest that lasted was in an American consulate where the (Polish) person in charge mistook me for a Pole trying to get a visa instead of an American needing information for my visa to stay in Poland. I've usually been mistaken for German(?) or Czech(!) when they realize I'm not Polish. No one's ever realized I was American (with no Polish roots at all!) until I tell them. I don't have an American r at all in Polish and most of my mistakes are plausible and occasionally the kind that Polish people use. The biggest tipoff is my general inability to produce the Polish l in final position I usually say angiełski (in old-fashioned stage pronunciation). and I make my fair share of mistakes with sz ś (and other similar pairs so that I say I'm combin my hair when I mean I'm glad).
I've also been surprised how many (educated) Poles can tell where Polish people are from by their speech alone. The standard has almost completely replaced dialects. The most that most Polish people can discern is whether a speaker is from a city or the countryside IME.
English people can usually easily recognise the origins of non native and even native speakers ( to within approx 50km with native speakers ).
Americans can't. The dialect situation is completely different and most Americans have trouble placing people by accent. One reason is the rise of the SAE "from nowhere" accent. But IME Americans fail in locating people by accent in all but the most exaggerated, stereotypical cases.
I'm hopeless with British accents, I can tell RP-ish (or Estuary) from other kinds, can vaguely tell the difference between Scotland and everything else and sometimes between Northern and Southern English but markers that are crystal clear for Brits mean nothing to me.