Answer:
Yes (and no!)
The styrene molecule is a benzene ring with a -CH=CH2 group attached. In the polymerisation process, the double bonds in two adjacent groups are broken and joined to form a chain:
-CH-CH2-CH-CH2-CH-CH2-CH-CH2-....... with a benzene ring off each -CH- group.
I believe the benzene group is referred to as a "phenyl group" in complex molecules, such as polymers like polystyrene.