No. the blood that is shed during menstruation is actually a surface layer of endometrium that builds up to prepare the wall of the uterus for implantation of a fertilized egg (ovum), in the event that fertilization occurs. If fertilization doesn't take place (no pregnancy), then in humans that surface tissue is shed to the body exterior, called menstruation. In horses, the body absorbs the tissue into the bloodstream, so menstruation doesn't occur. Estrus just refers to the cyclic release of ova (eggs) from ovaries, which is usually referred to as ovulation in humans, rather than estrus.
Starlight 1.....The pineal gland is not a special gland found specifically in the brains of horses. Humans and other mammals all have pineal glands. Hormones like melatonin are not only synthesized in the pineal cells, but also in red blood cells, peripheral nerve synapses, cells of the GI tract, and various other cells of the body. Receptors for reproductive hormones are also found in pineal cells, meaning that the pineal also responds to the presence of these hormones in blood. The pineal cell types vary among mammals, and the response to light energy is only one factor in many that influence the synthesis and release of hormones and multipe other chemical substances synthesized by pineal cells. There is still quite a bit that we don't understand regarding the role of the pineal gland in regulation of estrus cycles in horses and menstrual cycles in humans, and there is no one size fits all answer to all questions regarding these cycles.
Because horses are prey animals, they absorb their endometrial slough so it doesn't attract predators as it would if they menstruated to the external environment. Horses still cyclically develop the endometrial tissue just as humans do, and it is eliminated when pregnancy doesn't occur just as in humans. They just don't menstruate.