The source is in the Talmud, Megillah 31a (link). It is a statement of Rav Huna, in the name of Rav:
אמר רב הונא אמר רב שבת שחל להיות בחולו
של מועד בין בפסח בין בסוכות מקרא קרינן
ראה אתה
Rashi explains:
מקרא קרינן ראה אתה - שיש שם מצות שבת
ורגלים וחולו של מועד דכתיב את חג המצות
תשמור ומכאן למדנו איסור מלאכת חולו של
מועד במסכת חגיגה (דף יח.):
My free translation: The portion we read is "See, You" (Shmot 33:12) -- Which has in it the mitzvot of Shabbat, and the pilgrimage festivals, and chol hamo`ed, as it is written (34:18): "Keep the festival of Matzot," and from this we learn the prohibition of work of chol hamo`ed, in Tractate Chagiga (18a).
So, it's not a question of "duplication," exactly; rather, this passage mentions Shabbat, the holidays, and the pasuk which is the derivation of chol hamo`ed's prohibition of work. This section of the Torah is simply the best fit for Shabbat chol hamo`ed, whether during Pesach or Sukkot.
[Now, you might say, well, according to that Gemara in Chagiga, there are three (or five) opinions for the source of prohibiting work on chol hamo`ed. Why not read one of the other sources? I don't have an answer for that.]