Correct, "airplane" is U.S. English while "aeroplane" is U.K. English. Both terms can be shortened to "plane".
For the information of non-English speakers, "plane" has other meanings, so context is important.
Duo teaches American (U.S.) English and often accepts U.K. and other spellings (so report them so they can be added to the list of correct responses).
Can someone clarify the differences between fly, flyvetur, and flyvemaskin? I keep an excel file in which I log new vocabulary as it's introduced and though I have occasionally been known to make mistakes in doing this, I have logged both fly and flyvemaskin as airplane. Also at some point I noticed in one of your exercises you translated fly as flight but you have also introduced the word flyvetur for flight.
Flyvemaskine refers to an airplane (lit. "machine for flying"), and fly is the short form of that, like "plane" in English. Flyvetur is a trip through the air, typically in a plane.
I'm not sure where fly was translated as "flight", but it might have been due to a difference in grammar between English and Danish.