Thank you for the fun question. I have done some analyses and counted 123,029 rivers in the Andes. To arrive at this number, I used the global Free Flowing Rivers (FFR) river network, a related HydroSHEDS product, created by Grill et al. (2019). The FFR river network uses the same discharge and catchment area criteria as HydroSHEDS to define a river. To define the Andes, I used an ecoregion classification dataset that showed good agreement with topographic data and the definition you provided. To offer a comparison between the number of rivers in the Andes and at the global scale, I also counted the total number of rivers in the global FFR river network, which was 4,368,233.
Grill, G., Lehner, B., Thieme, M., Geenen, B., Tickner, D., Antonelli, F., Babu, S., Borrelli, P., Cheng, L., Crochetiere, H., Ehalt Macedo, H., Filgueiras, R., Goichot, M., Higgins, J., Hogan, Z., Lip, B., McClain, M.E., Meng, J., Mulligan, M., Nilsson, C., Olden, J.D., Opperman, J.J., Petry, P., Reidy Liermann, C., Saenz, L., Salinas-Rodriguez, S., Schelle, P., Schmitt, R.J.P., Snider, J., Tan, F., Tockner, K., Valdujo, P.H., van Soesbergen, A., Zarfl, C. (2019) Mapping the world’s free-flowing rivers. Nature 569, 215-221.