What is Jewishness? Don't get stuck in semantics. It's not
'someone who is connected to the tribe of Judah, because the etymology of "jew" is "yehuda".'
Jews are those who Hashem chose for the special mission of receiving the Torah and being a light to the nations. This started with Avraham, and passed to one of his children, Yitzchak, and then passed to one of Yitzchak's children, Yaacov. See here for Scriptural reference to this process.
This was then passed to ALL of his children, meaning that every child of Yaacov received this mission. Indeed, all of the people camped around Sinai were the children of Yaacov, representantives from all of his own children (the 12 tribes - also there were converts and eruv rav but that is beyond scope).
Once the Torah was given, we were obligated in all of its commandments, as well as the tradition given along with the Torah which contains the explanation and methodology to the commandments. It is from this that we understood that "choseness for 'Judaism'" passes through the mother.
So, tl'dr, don't get stuck on the etymology - "Jewishness" is more connected to the original Jews, Avraham, Yitzchak and Israel (Yaacov). All the descendants of Israel, through matrilineal heritage, are "Jewish", i.e. of the House of Israel who were chosen to receive the Torah and be a light unto the nations.