For example, people who discover to have Jewish descent and begin feeling themselves as being Jewish as well, because the feeling of "brotherhood" towards other Jews is strong. Is it a Jew, semi-Jew, absolutely not a Jew or is there a specific term for it?
According to traditional Jewish Law (Halacha), a person is considered a Jew if his mother is Jewish. Therefore, if his mother's mother's mother's mother was Jewish, then he himself is also Jewish, even if all of those female ancestors married non-Jews. On the other hand, if even one person in the chain of mothers was not Jewish, then the person himself is not Jewish, even if they all had "Jewish" families and married Jews.
Self-identity has absolutely nothing to do with a person's Halachic Jewish status. Just because someone does not feel Jewish does not mean that he is not Jewish and bound by the laws of Halacha. On the other hand, someone who "feels Jewish" but who does not have a Jewish mother is not Jewish according to Jewish Law (unless s\he goes through a formal conversion process).
Another argument would be that the term Jew and Jewish is connoted as being a race which despite much inter-marriage between Jewish communities, still doesn't justify the term race. It is a belonging to a group of people that is widely encouraged to support and justify the need for the state of Israel. Therefore, if being Jewish isn't a race then quite simply any individual who wants to practice the Jewish faith should be a Jew, the same way people convert to any religion. Of course the Jewish establishment and clergy would vehemently disagree but it's a fact that Jews are not a race.