The claim to a caliphate by ISIL has little to do with the caliphate as an historical unit; it died with the sack of Baghdad and the removal of the caliphate to Cairo, where it became a tool of the Mamluk rulers.

ISIL claims to be the successor to the political and religious community established by the Prophet Muhammad. The ideal of a united church and state leadership is an old one, and desired by many Muslims.
With large areas in Syria and Iraq, ISIL has effectively eliminated the boundary between the two areas.

Muslim beliefs at present range from moderate to rigid. Shiites believe that leadership should have followed through descent from Ali; creating an early split. It is doubtful that the groups will get together, especially with the jihadist elements in ISIL.