The only way to become the monarch, other than dealing damage to an existing monarch, is to play any of the cards that state “you become the monarch” like Thorn of the Black Rose or Crown-Hunter Hireling. Whenever a creature deals combat damage to you, its controller becomes the monarch.” So as long as you’re the monarch, you’re gonna be drawing an extra card until you take combat damage.
To put it simply, that means that it’s as if you gained a “player type.” Once a player becomes the monarch, they gain a status that states: “ At the beginning of your end step, draw a card. Monarch is what’s known as a player designation. Garrulous Sycophant | Illustration by Lake Hurwitz