First, it was Omega who lost his cool but now, it was Kahn himself who was angered and openly challenged the Hero of Space for a fight.
Why did Kahn have such an intense reaction?
Omega losing his composure suddenly over the intervention by the hero of space was not anyone expected, probably not even Omega himself.
However, whether it was the beginning or the ending… Kahn too felt that Ryuken was being a hypocrite.
Kahn said that he'd like to see how far the hero's companions could withstand Omega/Raiden's onslaught.
But he never stopped or deterred Ryuken from acting on his own and intervening in the battle when his fellow members used the Sanguine Formation of Death just for the sake of winning, even though he too knew that Omega could almost certainly die despite this only being friendly duel as per his own words.