Is there such a thing as a "long haired" horse?

Not the way there are long haired cats or dogs. Some breeds grow longer and shaggier winter coats than others, but pretty much all horses have fairly short fur.

However, there IS a long-haired breed of donkey, called the Poitou, which has dreadlocks similar to those seen on Puli dogs. The long-haired trait is dominant so it shows up in part breds. It does not, however, show up in mules bred from Poitou donkeys.

A Poitou donkey with a full coat. Image source Remi Jouan via Wikimedia commons.