Yup, looks like you've spotted it -- in particular, the specular term is
materialSpecularColor * lightStrength * pow(cosAlpha, 10)
That's the part that will contribute to the shininess of the material. And, materialSpecularColor comes from highp vec3 materialSpecularColor = lightColor.rgb; which is to say that the materialSpecularColor in that shader is not really a property of the material since the value is just from the lighting. Not really the clearest name if you are trying to pick it apart. This type of look is traditionally used for a plastic type material which has white highlights for white lights regardless of the color of the material. If you want something a bit more metallic looking, you can multiply the lightcolor.rgb by some actual material color. This lighting model is a bit of an old school aphysical hack, so you may want to look into to more modern physically based lighting models if you are looking for more realism. But the classic specular + diffuse model is way simpler to understand even if it isn't super accurate to the real world.
