Hand foot and mouth disease==is that like hoof and mouth in animals?

Very different. Although hfm is most often associated with coxsackievirus a16, the process does occur with other enteroviral strains (enterovirus 71 & others). A child will develop immunity to an individual strain after an illness but be susceptible to other unrelated strains, much like there are different strains that cause the flu. It has no relationship to the animal disease.
Both are viruses. Both are virus related.