Based on your rationale you can end-up with race discrimination and nazi stuff. (e.g even if you are born in the same country your parents were immigrants so you are still a kind of non-native and so on).
The cultural issue goes both ways. The wave of immigration may make the natives feel discriminated as well. That doesn't make them "immigrants".
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.
That aside my point is that it's wrong to put an immigrant on a visa to Germany on equal footing with an EU citizen. The former "may be deported" at any time for various reasons while the later has *all the rights german people have.
Discrimination happens even between different regions, the most common being in the U.S(i.e souther accents). There is also race discrimination even for "native" people. That doesn't make all the black, yellow people immigrants from Africa, Asia etc.