Is it possible to separate a layer's effects from its video?

Without any understanding of the project structure nobody can tell you where footage may need to be replaced or how effects can be re-used by ways of presets, copy&paste or whatever. At the very least you'd have to provide a link to the original clip/ project download or screenshots of the relevant timeline(s) with the properties twirled open.

Mylenium

What I can recommend in these cases is to ask the creator of the video how to do it, if this effect comes in an effects package it must have some kind of vendor support. We cannot tell you exactly how to do it because we have no idea what effect it will be and you do not show it to us in your question, surely you have to dig into the layers and their effects and properties to achieve what you want.