Why would a time signal receiving watch have an airplane mode?
Most useful radio receiver designs utilize some form of the superheterodyne architecture, where one or more radio-frequency local oscillator signals are used to shift the frequency of the received signal to an intermediate frequency where it is more readily processed.

Typically, the local oscillator signal will radiate from a receiver to some degree - how much depends on the specific design, but it is not uncommonly detectable in near proximity.

Restrictions on radio frequency sources on passenger airlines have historically tended to err on the side of caution, restricting anything that could theoretically emit moreso than being specifically tailored to demonstrated concerns.  Since a local oscillator theoretically radiates, the argument was that you should not be operating one, which effectively meant that you should not be operating a radio receiver.

Of course digital processing circuitry typically radiates clocks and their harmonics, too...  But then, the shift in the last several of years towards services like in-flight WiFi also indicates a shift in thinking away from vague theoretical concerns.