Can I have a layer style apply to multiple layers at the same time?

Depending on how your content is set up, you could instead apply a Solid Color fill layer (Layer > New Fill Layer > Solid Color or go to the bottom of the Layers panel, click the middle icon, and select Solid Color...) above your layers, give the fill layer a blend mode, and apply a Layer mask to the fill layer to exclude the middle layer. Then change the color of the fill layer whenever you want.

To apply a layer style to multiple layers simultaneously, you could either create a Smart Object using the layers or group the layers and apply the layer style to the Smart Object or the group. The layer style would affect a layer in between them though.