To become a historian (i.e. by completing a historical education to the level of a PhD), you would be expected to focus on a single field of specialization. That is to say, you could be a scholar whose field of specialization is medieval Europe (with your particular topic(s) of expertise being some narrow subset of that). However, in the course of your education, you would be expected by almost all graduate programs to acquire a broad familiarity with the historiography of other time periods and regions as well. This would serve two primary purposes. The first purpose is for teaching - you would probably be expected to be able to teach survey courses on European history or "Western Civilization" as a whole. The second purpose would be to make you a well-rounded scholar. If you were to focus solely on Medieval Europe and not educate yourself on other geographical regions and time periods, you would be unable to place your field of interest in any wider context, and would be stuck with a narrow, parochial view of history. That being said, your study of other periods and regions outside of your field of specialization would naturally be in less detail.

So to give the example of my own education: My field of expertise is the early modern Ottoman Empire. That's what I'm training to become an expert on, but I also need to be educated in Islamic history as a whole, from the origins of Islam to modern times; I need to be educated in early modern European history, as otherwise I wouldn't be able to place the Ottomans in context with their western neighbors; and I need to be educated in the field of world history, particularly in the early modern period but more broadly as well.

To put it in simple terms, it may be helpful to think of your specialization as radiating outward from your topic of interest chronologically and spatially. You should be familiar with your region of expertise across broad expanses of time, and you should be familiar with your time period of expertise across a broad geographic space. So if you specialize in medieval Europe, you should be generally familiar with all of European history, and broadly familiar with the history of Europe's neighbors during the medieval era, for instance the Islamic world. However, you would probably not be expected to know much about, say, ancient China or the modern Middle East.

However, there are certainly alternate ways one could choose to focus one's education - for instance, if your topic of interest is women and gender in medieval Europe, it may be more useful to study the historiography of women and gender more broadly (i.e. not tied to a particular period or region) rather than adopting the framework I described in the above paragraph. Some graduate programs are more flexible than others, but most will certainly give you leeway to pursue the topics that are most relevant to you.

As a current college student majoring in History, you have a lot of options when it comes to classes that you get to pick. For example, all of my classes that I have chosen are American History based. Now that being said, I have studied everything from Native Americans to African Americans, and everything in between. I'm also Minoring in African American culture. So in all, you get to choose, but be ready for slightly less interesting topics involving your Major. However, taking all that into account as a College Graduate with a History Degree, you should be able to recite history from all time periods, and constantly strive to learn more and more about topics. So yes, you may find one topic to be your motivation for studying history. Yet you should always be ready and willing to learn about others.

/u/Chamboz has already given a really excellent answer, so I will be brief.

As you advance from undergraduate to graduate work, assuming that's what you mean by "real historian", your specialties will become increasingly narrow. An undergraduate degree in history will allow you to be broad and to learn a little about a lot of things. Moving into an MA degree, you will begin to focus more - perhaps now on European history or Medieval Studies. Doctoral coursework is similar, focusing perhaps more narrowly still (depending upon your department's resources) on your area of specialization. Finally, the dissertation will be the most niche thing you've ever done. You will be an expert on the most narrow aspect of Medieval history. Your training, however, will allow you to teach a much wider array of classes than your dissertation topic would indicate.

I'm not in a history department, but a theology department. My work is historical, however. I'll say that if you want to focus on Medieval history, you will not be to do so and to do a good job unless you understand, at least broadly, its antecedents. From the "rediscovery" of Augustine in the 9th century to the beginning of the Renaissance, there is a recovery and interpretation of antiquity always at work in Medieval authors. It's hard to appreciate Aquinas without knowing something of Augustine, Ps. Dionysius, Aristotle, etc. In order to appreciate your subjects, you must come to appreciate the things they appreciate.