If you have ffprobe installed:
ffprobe -loglevel quiet -show_streams -show_entries stream=index -of csv <input> | wc -l
This will output one line per stream, and count the number of lines with wc. You can also filter for video or audio streams by adding -select_streams v or -select_streams a, respectively. See the manual for more details.
If you do not have ffprobe, ffmpeg can be used too, but in a slightly less robust way (i.e., this may break if the output format changes or something contains the words "Stream #" in the output):
ffmpeg -i <input> 2>&1 | grep "Stream #" | wc -l