I think the answer is
Solids only
He likes toast but not cereal.
Both breakfasts, but he likes the one with no liquid - milk.
He likes bicycles but not motorbikes.
Both bikes, but one without liquid - gasoline.
He likes ice but not water.
Ice is the solid form of a liquid - water.
He likes lamps but not lava lamps.
Both lamps, but he likes the one with no liquid - water mixed with mineral oils + liquid carbon.
He likes claw hammers but not spirit levels.
Both construction tools, but he likes the one with no liquid - ethanol.