DISCAIMER: The following is just my understanding of the issue based on my study. For this and any other halacha l’maaseh issue, consult your own Rav or RH.
It would seem to me that any cheese, like Swiss cheese, that uses a coagulating agent requires hashgacha. The gezeira on gevinas akum is separate from the gezeirah on chalav akum. Although the reason for the gezeira isn’t really clear from Shas and Rishonim, the general consensus is that the rennet used as a coagulating agent is the concern and hashgacha must ascertain that it comes from a kosher animal. In fact, almost all commercial cheese made in the U.S. use microbial rennet (grown from yeast) instead of animal rennet. Nevertheless, most contemporary poskim opine that hard cheese needs a hechsher.
Soft cheeses such s farmer cheese, cottage cheese and cream cheese generally don’t use a coagulating agent at all or use vinegar. As such, many poskim hold that such cheeses do not need a hechsher.
N.B. The reason that many are meikil in this issue is the fact that true cheese only be made from the milk of a kosher animal. The preceding, however, does not apply to “processed cheese foods” and one must be wary.