"Inheritor of Death, you really intend to take the side of a stranger over your loved one?" Agatha felt like her pride was wounded upon seeing Azzy rescue Estella once again. If not for the fact that she sensed the presence of the Reaper, she wouldn't have hesitated to attack him for being preposterous.
Usually, one's Arcana spirit influences their host's personality a lot. While Claire didn't inherit any arrogance or pride, she still received her violent/aggressive side from Agatha.
To her accusation, Azzy calmly responded with two statements, "First of all, you aren't my loved one, Claire is. This action of yours is independent of her decision. And second, even if it's Claire, I would still do the same thing because I'm the referee here. A referee needs to be impartial at all times."