Cause "seeing a horse" may have meant you were going to
rustle one which in certain parts of the country may have been
a capital offense. So it was better to say you were going
to see some one (certainly not a woman) about a horse. Cause
if you were going to see a woman about a horse, you may have
meant you were going to "horse around" with the woman!
Anyway, it's just my poor etymological guess.
Perhaps simply because 1 could actually meet with a man to buy a horse. "Seeing a horse" doesn't really mean anything but literally seeing a horse (looking at a horse), which isn't as much action as meeting a man somewhere.