My question is how did the study of politics become systematized and institutionalized from political theory (eg Aristotle, Hobbes) and practical statecraft (eg Machiavelli) into modern (ie post-1800) political science, as well as how it came to include major defined and widely-accepted subfields like comparative politics, international relations, political theory, etc. I know that by the early 1900s, it's become an established university department such that academics like Woodrow Wilson and Stephen Leacock are making a living as tenured professors of political science. But how did we get to this point?
Likewise, I'm struck by how the discipline seems to lack a historical understanding of its development, considering other social science disciplines often tell imagined/mythologized narratives of their modern development (at least for your standard 101 course), identifying crucial thinkers as inflection points towards a more rigorous, scientific study. For instance, sociologists point to Marx, Durkheim and Weber; anthropologists to Boas and Malinowski; even historians point to von Ranke as a critical pioneer of the modern historical method. Why hasn't political science developed its own disciplinary identity in the same way that anthropology, sociology, economics, history, etc have?
Short answer:
"how did the study of politics become systematized and institutionalized"
It didn't.
Political Science is no single discipline, and so there's no single narrative that explains it. Its different things to different people and different things in different institutions. It continues to evolve, and to the extent that it has founders -- writers like Thucydides, Machiavelli and Vico, they don't belong to the discipline, and they're only relevant to some of its practitioners. It's a relatively new discipline, and just what it is depends very much on the specific institutional history.
Discussion:
Funny you should mention Woodrow Wilson: Princeton doesn't have a department of "Political Science" -- its the "Department of Politics" and the omission of "science" was Wilson's own decision. Similarly at Harvard, it's the Department of Government. At British Universities, you'll find that its often called PPE (Politics and Political Economy), so even the name of the discipline is uncertain.
Just looking at the discipline as it developed at Princeton, what it was very much depended on who was running the program. So we can speak of Lyman Atwater a philosopher with a strong theological background who taught in the 19th century; he was a professor of "logic metaphysics ethics, ethics, economics, and politics". Or we can speak of Edward Corwin-- invited to join the faculty by Wilson, he was appointed a professor of Jurisprudence . . . and later was President of the American Political Science Association. The only real through line to this history is that of the institution itself, and perhaps the response of non-Marxists to the rise of Marxism in the academy.
Very generally speaking, the notion of there being a "science" to the study of politics, like the notion of there being a similar science to the study of history is closely associated with Marx (with an important nod to Giambattista Vico) and many institutions eschewed that suggestion. They also were often skeptical not just of the association, but the direction. Nonetheless even non-Marxists had to respond to the emerging intellectual challenge of Marxism, but the non-Marxist "Political Science" remained a bit of a grab bag.
"Modern political science" isn't any one thing-- its a remarkably heterogeneous discipline, ranging from traditional studies of government and empire that draw substantially on Thucydides (think Gilpin's Theory of Hegemonic Stability), to very data intensive study of population cohorts, their economic and educational attainment and their voting behavior (think Butler & Stokes' "Political Change in Britain"). Or they might be quite theoretical and overlap with philosophy. Philosophers like, say, Princeton's Philip Pettit or the late John Rawls are/were not "scientific" in their aspirations, and they share very little common ground with, say, Donald Stokes or Yale's Bruce Russett.
If you read the American Political Science Review today, it's often very hard to say "why this is Political Science rather than economics, epidemiology or sociology" -- and indeed you find academics from these disciplines writing on topics and with tools that are substantially similar. Perhaps the most consistent marker for "this will be a political science paper" will be quantitative study of voting or regime behavior, but that's just a subset of what you find in the discipline.
By the same token, political philosophers will read writers dating back to Thucydides-- Graham Allison put "the Thucydides Trap" into the newspapers, but that's an article that could easily have been written by a classicist. Indeed one often finds classists writing works that are or aspire to be contemporary political science-- think Victor Davis Hanson or Walter Scheidel. And the mathematicians and economists who gave us "the Prisoner's' Dilemma" and game theory . . . much APSA ink spilled on these subjects, but what makes it "political science" rather than something else is not easy to say.
So it's a bit like anthropology in that someone saying "I'm an anthropologist" only restricts but does not define what they do-- it might be recovering ancient DNA or interviewing gang members in Albuquerque. Similarly a political scientist might be studying voting behavior, campaign contributions, how wars begin or end, international organizations, international trade or many other quite unrelated subjects.