When Knights of the Black Rose enters the battlefield, you become the monarch.
Whenever an opponent becomes the monarch, if you were the monarch as the turn began, that player loses 2 life and you gain 2 life.
Abilities that trigger whenever you “become the monarch” trigger only if you aren't already the monarch. For example, if you are already the monarch as Custodi Lich enters the battlefield, its last ability won't trigger.
Being the monarch carries two inherent triggered abilities. “At the beginning of the monarch's end step, that player draws a card” and “Whenever a creature deals combat damage to the monarch, its controller becomes the monarch.”
If the triggered ability that causes the monarch to draw a card goes on the stack, and a different player becomes the monarch before that ability resolves, the first player will still draw the card.
The game starts with no monarch. Once an effect makes one player the monarch, the game will have exactly one monarch from that point forward.
The last ability of Knights of the Black Rose will trigger even if the most recent monarch was another player, as long as you were monarch as the turn began.