When people used to wear tefilin every day, did they do so on Shabbat as well? If so, why did people stop wearing tefilin on Shabbat? If not, why not?
Menachos 36b - Rabbi Yosi & Rabbi Akiva say that it is forbidden to put on Tefilin on Shabbos. The reason according to Rabbi Akiva is that Shabbos is instead the sign of Tefilin.
Please see here regarding the need of having two signs every day of our Jewishness. The two signs are Bris & Tefilin. On Shabbos since Shabbos is a sign, Tefilin is not necessary.