Can you equip multiple versions of the same module i.e. three General Research Pods? If so, what advantages would it give me to do so?

Yes you can, except where a specific limit is noted in the announcement post (for example, only one telescope array can be mounted despite the fact that it's a 2x1 module.) The impact of multiple modules of the same type would depend very much on the function of those modules, but in terms of the base modules it usually means you can conduct more research or analyze more data with the separate modules working in tandem. (Of course, installing extra fuel modules would have a more immediate impact: doubling or tripling your extra fuel supply!)