Why is styrene an aromatic compound?
As Wikpedia Article says ,the criteria for a molecule to be aromatic are --


the molecule must follow Hückel's rule, having 4n+2 electrons in the delocalized, conjugated p-orbital cloud;
the molecule must be able to be planar;
the molecule must be cyclic; and,
every atom in the ring must be able to participate in delocalizing the electrons by having a p-orbital or an unshared pair of electrons.


When we are talking about aromaticity , we consider the electrons inside the ring. You are right that the styrene has 8 pi electrons but when checking for Huckel's rule, we only consider the electrons inside the ring. The alkenyl group is just an attachment to the stable aromatic compound which might take part in resonance but wouldn't affect the compound's aromaticity.