Almost the same rights but not quite. You are also confining this issue to a purely legal standpoint -- my ability to travel the EU freely does not mean that I am Italian, for example.
edit: after rereading your comment, perhaps this passage refers to after I hypothetically take my citizenship. To answer that, I would consider myself then both German and an immigrant.
The point is, I do not identify as German, and Germans do not think I am German. I am influenced by German culture of course, having lived here long enough, but I am not strongly enough influenced to call myself German. I have younger siblings who are born here and have no memories of our family's homeland. I think of them as German, since they are in every way so -- they speak German, have German mannerisms and ideas, and do not understand a single bit of the culture of where I was born. They do not have German citizenship, but that doesn't matter at all since it's just who they are.
I hope this clears up my feelings on this topic.