There is a theory of rainbows due to elliptical droplets. It was started by Willy Möbius  but is perhaps more clearly described in the modern papers [2,3]. Unfortunately the math is messy. While this may change the appearance of the rainbow, the biggest effect is on the angles between the different order rainbows: by measuring them one can presumably work backwards and get a value of the ellipticity.
Except for one problem: raindrops made ellipsoidal because of wind oscillate (they are water after all). They will hence change their ellipticity rapidly; to make matters worse, the degree of maximal ellipticity change scales roughly parabolically with droplet size and droplets can become non-ellipsoidal [4,5,6]. Hence a real rainstorm (with different droplet sizes and vibration modes) will produce a rather messy signal.
So the answer to your question is likely "yes, for rigid uniform droplets" and "no, not for real rain".
 Möbius, W. (1910). Zur Theorie des Regenbogens und ihrer experimentellen Prüfung. Annalen der Physik, 338(16), 1493-1558.
 Lock, J. A., & Können, G. P. (2017). Rainbows by elliptically deformed drops. I. Möbius shift for high-order rainbows. Applied Optics, 56(19), G88-G97. http://www.guntherkonnen.com/articles/322
 Können, G. P., & Lock, J. A. (2017). Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers. Applied Optics, 56(19), G98-G103. http://www.guntherkonnen.com/articles/323
 Szakáll, M., Mitra, S. K., Diehl, K., & Borrmann, S. (2010). Shapes and oscillations of falling raindrops—A review. Atmospheric research, 97(4), 416-425. http://e-science.sources.ru/sites/default/files/upload_forums_files/y0/sdarticle2.pdf
 Thurai, M., Bringi, V. N., Manić, A. B., Šekeljić, N. J., & Notaroš, B. M. (2014). Investigating raindrop shapes, oscillation modes, and implications for radio wave propagation. Radio Science, 49(10), 921-932.