There are horse breeds that grow mustaches!
Most common horse breeds are the gypsy vanners (also known as Irish cobs), but it's also found on other feathery breeds (such as the Shire).
How? Vanners have a gene that makes them have their beautiful thick mane, tail and feathers. The more of this gene a horse has, the thicker the mane, tail and feathers and thus, the bigger the mustache!
Why? The mustache is to differentiate different types of grass. This makes grazing easier in poor light conditions!
And yes, some owners shave the mustaches off. It doesn't harm the horse, or sometimes, the horse eats its own mustache when it's too long!
For the Irish cobs, it also depends on the type of cob. I have a grai-type (the skinnier version), and grai's don't have a lot of the gene, but the cob-type in our stable does have a small mustache (and yes, I do comb her mustache).
CONCLUSION: Irish cobs are distinguished gentlemen and -women!