In short, in actuality there is no real geographic distinction between Kuwait and Iraq, this much is clear. However, there exists a distinction in the sense of proximities and population factors. During the Abbasid caliphate and later periods, Shi’a and Shurha/Khawarij communities existed most strongly along the Euphrates River and nearby. Bedouin groups often varied in their religious affiliation but many of whom were Shi’a or Shurha and travelled far distances to and fro.
Kuwait on the other hand, is more along the periphery outside of this Shi’a radiation of Twelverism on the Euphrates. So there is a northern Shi’a radiation on the Euphrates and Tigris River valley in the northern sectors and among the types of communities existing therein. This is must strong on the Euphrates near Karbala, Kufa and Najaf. Meanwhile, the other Shi’a radiation is in the al-Haasa region to the south of Kuwait, in modern eastern Saudi Arabia and in the far southern reaches of the Nejd. This sector of Shi’a radiation, began with the group known as the Shumaytiyya in the 8th century. This group in turn, was related loosely to all the major Shi’a groups of the time. From these Shumaytiyya arose the Qarmatians and other Is’maili Shi’a who would be common in the Haasa and the Nejd. These communities subsequently would blend into a radiation of Twelver Shi’a Islam within the Haasa after the failure of the Qarmatians and the decline of the Fatimid caliphate. The Ismaili communities in the Nejd would disappear in that very early al-Saud monarchy within the Nejd, which positioned itself along Sunni lines.
Thus, as a very simple answer, Kuwait as an area, is somewhat between and distinct from these radiation zones and was sparsely populated. In the Middle Ages, it was known that Shurha and Shi’a tribes had influence there, however defeats of these movements, the Zanj rebellion and the decimation of the Qarmatians, would have disbanded any sort of gatherings there. Any non Sunni religious radiation that would have occurred, would have been most likely eradicated by the Saljuq empire. While other centres of Shi’a radiation were too dense, settled and or isolated to be disbanded.