Honestly I don’t see that many similarities between Liam and Arthur/Artura, at least, not yet.
Both are very devoted to their people and feel a strong sense of duty to do what’s right because their kingdom rests on their shoulders. Both are very kind and enamored with the main character of their respective stories, and both feel the leaden weight of the crown. However, that’s where the similarities end.
Liam is shown throughout the story to really feel the crown’s weight. He constantly worries about being a good king and even considers abdicating three times dependent on your relationship with him. He first considers abdicating when he’s stuck in the engagement with Madeleine, he briefly considers abdicating when Anton Severus and the Sons of Earth were committing acts of terrorism against Cordonia, and he tells the MC if she married him in TRH 3 that he’ll abdicate the throne and go on the run with their daughter if Barthelemy is crowned regent.
Arthur/Artura seem much more resigned to their role as monarch. From what we see in the first couple chapters, even though they don’t deny the heavy weight of the crown, they also enjoy the difference they get to make with their brothers and sisters-in-arms.
I would say Arthur/Artura is less like Liam and more like Hunter Fierro from TRM.