I guess "Carolingian" is a result of the predominantly English speaking WikiTree clientele, particularly early on. The modern Germanic ending would be -ing singular and -inger plural, and the C would be a K.
Historically the family did not have a name as such, that was given to them later, in the East Frankish realm it was introduced in 965 by Widukind von Corvey as "Karoli", in the West Frankish realm after 991 by Richer de Reims. That then developed in the literature in Latin via Karlenses, Karlingi, Karolini to Karolingi.
In the absence of a family name the use of a dynastic name such as this certainly has its justification. Whether it should be in English is another question.