A few plot similarities does not a “sameness” make. Logan’s run was about post-apocalyptic underground society that had implemented its own mythology and rituals in order to survive. The Island is about genetic engineering and cloning and the deception of the occupants in order to keep them in line (more like The Matrix in this regard). The Island was about “big business” where Logan’s Run was about “big ideas” and perhaps government.
Maybe the comparison between the two troubles me most because I really like one of the movies and really don’t like the other one.