It's not wrong in English, but it means something slightly different. Some examples: "Our dad gave my brother a book, and me too." would imply that both my brother and I received a book each from our father, whereas the sentence "Our dad gave my brother a book, and I also." would imply that both I and my father each gave my brother a book. What I'm not sure about is whether the sentence in Swahili implies the object form "me" or the agent/subject form "I", or either.
So, since this phrase doesn't have a verb, it would be ambiguous in that sense?
In English, for both meanings, you would use: "And me too." "And I also" is just bad English.
Well, I've met natives who insist there is a difference, and that it is important. I usually follow what you are saying, as the other one sounds a bit old fashioned to my ears. Now when I think about it, maybe they prefer "And I as well." Don't know if it is better.
Technically speaking, yes "I also" or "I as well" would be more proper than "me, too" for cases where it refers to the subject rather than the object of the sentence; however, nobody actually speaks that way anymore. Almost any first person reference that far removed from its verb will be "me", even if it technically refers to a subject. Although, there are also times that people hypercorrect and say "I" when they really mean "me" as well. That usually happens in constructions such as "(person) and I." "Me, too" has simply become a set phrase in English.