This question assumes that the steed is acting independently.
Is the Steed responsible for maintaining the concentration requirements for a shared Aura of Vitality spell?
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As asked and answered in How does the spell "Find Steed" work with Aura of Vitality?, the steed can use its bonus action irrespective of if it is acting independently or not. A controlled steed has its actions limited, its bonus actions are unaffected.
As for concentration, it's the paladin's spell so only the paladin's concentration matters.