PBA: Sorry for the delay.
When a davar is pogem a ta’aruves, Shulchan Aruch paskens like the Rashba that the ta’aruves is mutar if the majority is heter. The basis for this Halacha is learned in the Gemara from neveila sheina reuya l’ger. This would certainly seem to be the case with our soap, and the ikar hadin should be that soap is mutar.
However, the Rambam writes that neveila serucha sheina reuya l’ger is patur, which implies assur. The Pri Chadash understands the issur to be bal teshaktzu, which would not apply in a ta’aruves. However, the Pri Toar understands that the issur neveila remains. The Chavos Da’as writes that according to the Rambam who explains the heter of neveila serucha is from “shelo k’derech achilaso,” the cheftza is still assur and the heter is on the ma’aseh achila.
Therefore, according to the Pri Toar and Chavos Da’as, while the soap itself is mutar, the non-kosher ingredients in the soap would still be assur. Based on this, it is understandable why many people would want to be machmir to use kosher soap, to avoid putting non-kosher ingredients on their dishes, and by extension, food.
Furthermore, the Teshuvos HaRashba, brought in Shulchan Aruch YD 134 writes that any ingredient intentionally added to a ta’aruves is never batel, no matter how miniscule. This would presumably apply even to a davar pagum that requires bitul b’rov according to the Rashba and SA.
Additionally, the Ran holds that rov heter is not enough. The con of the ta’am pagum must outweigh the pro of the added volume. While SA in YD 103 only brings this opinion as a yesh omrim, according to the Pri Megadim and Rabbi Akiva Eiger, the Shulchan Aruch paskens this way in Hilchos Pesach. This may be an issue with soaps if the non-kosher ingredient is critical to its cleaning performance.