Are there characteristics of photographic film that would make it an analog medium?
Film isn't absolutely "analog", as in continuous.  Every individual silver halide molecule, after exposure and development, is either metalized or not; and there are a finite number of these molecules in every frame of film, thus quantizing the exposure measurement.  However the density and location of the film grains and silver halide molecules is semi-randomized, which helps noise shape or dither the sampling, quantization and aliasing noise more than if they were in a regular grid with fixed step values.

So some current differences are not that film is in some sense a truly continuous measurement system (at least down to the Planck level), but more might be that the "sampling" is dithered, and also that the dynamic range may be higher than in the most common digital formats (8 bit integer per channel).