I was kind of thinking that it was town of Franklin, Idaho, but it is clear as a bell that the census record is for "Franklin" in Cache County, Utah Territory. So I took a look at a couple of crude maps of the Utah and Idaho Territories and from what I can see the border between the two was set when the Idaho Territory was formed in 1863, and never changed. There were changes made in the territory's borders, but it was caused by cutting out the Montana and Dakota Territories in 1864 and the Wyoming Territory in 1868. So I would say that for some reason, like, the people living there wanted to believe that Franklin was in the Utah Territory and used that on all official documents. Than at some point in time it was shown that it was in fact in Idaho and all documents where then shifted to that. This is kind of like the border dispute between Massachusetts and Rhode Island and why we now have a Bristol County in both.
Thanks for your help.