An illusion is not a creature. The rule is clear that Bobby must attack or take damage to an enemy creature to maintain his anger.
Spirit of the rule
If Bobby doesn't know that what he's attacking is an illusion and not an enemy creature, how does that differ from Bobby's view of actually attacking an enemy creature? If he learns that what he just attacked is not what he attacked, it should increase Bobby's anger. As a DM, I would say that attacking an illusion that the barbarian considers an enemy creature is enough to maintain the barbarian's anger.