Wikipedia, in the article "divisor", defines a fractional ideal sheaf to be a sub-$\mathscr O_X$-module of the sheaf of rational functions $M_X$.
However, Kempf's book Algebraic Varieties defines in page 63 the same notion with the additional requirement that the sheaf be coherent. Is that a true difference? What does it stem from? And why does either of these definitions provide an analog for the usual invertible ideals, as in here?
A fractional ideal sheaf is a subsheaf of the sheaf of total quotient rings $\mathcal{K}_X$ which is a coherent $\mathcal{O}_X$-module. Among fractional ideal sheaves, (generalized) divisors are those which are nondegenerate and reflexive as $\mathcal{O}_X$-modules.
See this paper by Harstshorne for a detailed exposition:
R. Hartshorne, Generalized divisors on Gorenstein schemes, K-Theory v. 8, n. 3 (May, 1994): 287-339