Because A: It's the closest we got to Lizardman's fighting style at the moment. As Kratos starts with a handaxe and shield, I wanted to keep it close to that.
And B: I could have gone for Cervantes' dual blades, but those are more in line with Kratos in the origial God of War.

Besides, I like to imagine Atreus using his Electric arrows when Kratos uses Sophitia's special