If you mean what I think you mean, I think that's just a misunderstanding:
“This here is Kaz. She’s one of my oldest and very calm. Nothing fazes her anymore.”
A light-brown peregrine falcon sits rigidly on a perch, clicking her beak as Azlean begins to fasten the leash to his glove and then the field jess around Kaz’s anklet. Without needing to give a command, Kaz climbs onto Az’Lean’s glove and he holds the field jess between his fingers to make sure she stays there.
“This here is $playername,” Az’Lean says and holds his left arm with Kaz on it out in front of you. Pitch black eyes stare at up you and Kaz slowly tilts her head to the side.
The falcon's name is Kaz, and he is introducing the MC to the falcon in that sentence, not the other way around. Though I see why it might not be entirely clear. I might go and reword that sentence at some point.
I can totally imagine Az'Lean (if he's on good terms with MC) starting to grow even fonder of owls! It'd be difficult not to associate them with each other, I think.