The purpose: To give more meaning to the sentence and for learning purposes (I had the opportunity to remember that term and to practice it).
You can compare the extra mee to the annoying little er, which is an added little term that may be necessary in Dutch, but completely optional in English. Examples: De katten kwam eruit -> The cats came out (of it) / Wat er gebeurt -> What happened (there).
In this case, mee is optional in Dutch and English. :)
A bit baffled by the question, I'm afraid. I'm open to correction, but here's my understanding: "mee" has no direct equivalent in English, but can be loosely translated as something like: "with", or "along" (though other posts in this discussion suggest "along" wasn't accepted here). So it's irrelevant who's doing the taking: it might be "I", "you", "he", "she", "they", or even (somewhat less likely) "it". It's a separable verb, "meenemen", so the only part that changes depending on the subject (whoever's doing it) is "nemen". Ik neem, hij neemt, zij (pl.) nemen - whatever. The "mee" doesn't change, just as "along" doesn't change in English, if you use: "to take something along". "I take it along", "you take it along", "he/she/it takes it along" etc.
Edit: I've just realised that possibly you are confusing Dutch "me" and "mee". Not the same word - nothing to do with each other. "Me" is just the unstressed form of "mij". In English, "me". So yes, it only applies to the first person. You can't use "me" of they/he/she etc. - just like in English. "Mee", on the other hand, is the separable part of a separable verb, and loosely corresponds to "with", or "along", as described above. You can have "meebrengen" (to bring with/along) as well as "meenemen" (to take with/along). There may be other separable verbs with "mee" that I don't know.