A pet peeve of mine is when people say that the United States, France, or Russia (three very different republics but all with a strong presidency) are still in some way "monarchical," because the President, the head of state, has a lot of power. No. No no no no no. Monarchy is not about one man having a lot of power. Monarchy is about so many other things: Tradition, Inheritance (the existence of non-hereditary monarchies does not mean that inheritance should be dismissed as irrelevant: most elective monarchies have had some sort of hereditary component to the process), Sovereignty being nominally vested in a person (whether or not that person actually wields power) who did not normally seek the office and is separate from the political process, Titles and Terminology, Aesthetics and Philosophy.
How much power the head of state holds in practice is completely irrelevant to the question of whether a government is monarchical or republican. A Monarch can be "absolute," serving as both head of state and head of government, or "constitutional," with a stronger (but formally subordinate) elected head of government; both kinds of monarchies are real monarchies. A Republic can have a strong president (like the three examples mentioned) with either a relatively weak (France, Russia) or no (USA) prime minister, or a ceremonial president (which I personally think is pretty much the stupidest thing ever) and a strong prime minister (or chancellor), like Germany or Italy; both kinds of republics are real republics. I wish everyone who writes about these topics could at least agree on terminology.