The story is that Fermi dropped the paper on the ground, not out of an airplane, and looked at the displacement of the paper as the shockwave passed to get a rough order of magnitude estimate for the shock wave's power, which could be then used to estimate the power of the explosion. I don't think it's apocryphal — he wrote about it at the time. You can read his brief report. The relevant section is at the end:

About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during and after the passage of the blast wave. Since at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.

I see no reason to doubt his account is genuine. That he dropped the paper was noted by at least one military observer with him, who mentioned it in his report on the test.