Well, he did propose that the President serve for life, which could be seen as monarchical.
Nobody had any idea at the time exactly how to set up a stable republic. Hadn’t been done for almost 2000 years. The closest was the Dutch Republic, and it had a lifetime executive.
Hamilton thought a stronger executive was needed than Jefferson or Madison did. Turned out they were right and Hamilton was wrong. But that didn’t make him a monarchist. He just thought the strong executive was necessary to preserve the Republic against its enemies, foreign and domestic.
BTW, the President’s present position in the national government is a hell of a lot closer to what Hamilton thought was needed than to anything Jefferson or Madison would recognize.
He was only speaking of the functional role of the Executive.
He never proposed a nobility or inherited Executive and repeatedly said that a monarchy was foreign to the spirit of the American people. Jeffersonian slanders made the ridiculous claim that H. was a monarchist. His press liars ran with it.