I knew the reason for this, but it took me some time to find an actual source for it.
The Halacha to take new fruits in the second night is recorded in Shulchan Aruch OC 600:2:
בקידוש ליל שני (ילבש בגד חדש) (הגהות מיימוני פ' כ"ט מהל' שבת) או מניח פרי חדש ואומרים שהחיינו ואם אין מצוי (בגד חדש או) פרי חדש עם כל זה יאמר שהחיינו:
At Kiddush of the second night (Rema - he should wear new clothes or) take a new fruit, and we say Shehechiyanu. If there is no (new clothing or) new fruit, nevertheless he says Shehechiyanu.
Be’er Heitiv records the reason:
חדש. ויש טועים ליקח אף בשאר י''ט וטעות הוא דר''ה קדושה אחת היא.
Some make the mistake to take even on other Yamim Tovim. It’s a mistake, for Rosh HaShanah is one Kedushah.
For background, to explain the concept of “one Kedushah”: An egg laid on Yom Tov is forbidden until after Yom Tov (Mishnah, Beitzah 2a). The Gemara (4b-5b) asks whether this extends to two days of Yom Tov and two days of Rosh HaShanah. The terminology used is Kedushah Achas, literally “one holiness,” versus Shtei Kedushos, literally “two holinesses.” The idea is that if it’s one long day, one “holiness,” the egg laid on the first day is forbidden on the second day, while if it’s two separate days, two “holinesses,” the egg laid on the first day is permitted on the second day. The Halacha follows Rav, that two days of Yom Tov are two “holinesses,” but two days of Rosh HaShanah are one “holiness.”
There are other ramifications to this Halacha besides for eggs. We make a Shehechiyanu on the first day of Yom Tov, but not on subsequent days. Outside of Eretz Yisrael, we make the Beracha on both the first and second day of Yom Tov, as the second day is treated exactly the same as the first (SA OC 496:1), but Rosh HaShanah is considered one long day (explicitly paskened by the Shulchan Aruch in OC 496:2). As such, there is a concern that, rather than making Shehechiyanu on consecutive days, one is making Shehechiyanu twice on the same day. Therefore, the custom is to be stringent and make the Shehechiyanu on something else, though we’re not worried if there’s nothing to make a Shehechiyanu on.