In Shulchan Aruch, (O.Ch. 131) פסח שני, is not enumerated as a day when תחנון is not said. The Pri Megodim explains Since פסח שני is a Yom Tov of Yechidim (individuals who couldn't make it on the 14th of Nissan) The same as the 9 days in the year when individual Kohanim brought Korbanos (Ta'anis 26a) yet we say תחנון since it's not the Tzibur celebrating, so too on פסח שני
it's not a YomTov for the Tzibur, תחנון is said.
There are many Poskim (סידור של"ה אלי' רבה סידור התניא א"א מבוטשאטש ועוד) who argue and hold not to say תחנון on פסח שני
Misnagdims complaint against Chasidim was that they don't follow Halacha. One of the examples they used was not saying תחנון on פסח שני.
They didn't complain on not saying תחנון on a Yahrzeit of a Tzadik. (which is a non-Halachic Minhag) Since this Chasidic Minhag of not saying תחנון on the Yahrzeit of a Tzadik is exclusively for Tzadikim Talmidei Bal Shem, therefore, this Minhag didn't start until after the second or third generation of Chasidim. (Unheard of before)
It is interesting, today except for Talmidei Chazon Ish, almost all Yeshivishe & Ashkenaz Minyonim don't say תחנון on פסח שני.