Foucault is normally considered a sociologist or social theorist, although it is important to note that many sociologists have written historical studies (in my field, for instance, William Rothstein is a sociologist who writes on the historical development of the medical profession in the United States.) Foucault is a historian in the sense that he is interested in tracing the development of societal institutions over time: though he isn't as interested in developing specific arguments about particular historical moments or contexts, and instead tends to write more sweeping narratives about what he calls "epistemes" or periods of thought. He's also garnered some criticism as being anachronistic, in that his work sells itself as a 'history of the present' or a way of understanding current society by looking at patterns in the past.

As far as his influence goes... yes, Foucault is hugely influential, especially in the social sciences, which some would argue includes history. He's the most frequently cited theorist in books in the humanities. In fact, my field (medical anthropology) is theoretically dominated by Foucault and similar theories of the power/knowledge axis, such that there is very little work done that cannot be traced in some way to Foucault, especially Birth of the Clinic. Part of this is because Foucault's work touches on so many subjects and his theories of power have wide applicability across many fields and disciplines, including a lot of postmodern gender theory work.