Does string theory predict a slightly different magnetic moment for charged elementary particles than QED does? I suspect there should be some variation if only because rather than particles being pointlike as they are in QED, they are instead 1-dimensional extended objects.
Of course, any such variation would be exceptionally small. Far beyond what we can currently measure.
Particles in string theory are not "extended objects". They are quantum states created by creation/annihilation operators just like in ordinary QFT, see also this answer of mine. No matter how often popular treatments of string theory get this wrong, string theory does not claim that point-like particles are "really" tiny strings.
Asking whether "string theory" predicts something is like asking whether "QFT" predicts something: Like QFT, string theory is a framework in which you can develop various specific theories (e.g. by choosing a specific compactification of one of the five string theories), but "string theory" in general does not make specific predictions about the particle content of the universe nor about any of its properties. QFT doesn't predict anything about the electron magnetic moment, it doesn't even "predict" electrons! The specific instantiation of QFT in the form of quantum electrodynamics, however, does.
The Born-Infeld action is different from the QED action so yes. String theory predicts that the magnetic moments of charged elementary particles will differ. I don't think there's any unique quantitative prediction yet though because of the usual problem with the huge number of string vaccua.
As for the comments, "0d particles" and "1d strings" are indeed both outdated concepts. QFTs which have earned a prominent place in the standard model necessarily admit a regime in which it is valid to ignore infinitely many couplings. These are the high dimension or non-renormalizable ones which should be there because of naturalness if nothing else. Although it is definitely more of a framework than a theory, string theory is now the following conjecture. Once you leave this special regime in which Feynman diagrams reign, you will find another one where the calculations are most easily done by appealing to effective strings. But worldlines of particles and worldsheets of strings should both be thought of as tricks for approximating a QFT.
The closest thing to a calculation that I can find, is in the 1994 paper "On the Anomalous Magnetic Moment in Heterotic Superstrings". This is a one-loop calculation of anomalous magnetic moment for "a massive spin-1/2 string state, with mass the order of the Planck scale, carrying a U(1) charge", in a model that can be supersymmetric or not, depending on parameters. They check that the outcome is zero for the supersymmetric case, as is predicted by some other results; for the non-supersymmetric case, they present the expressions that would need to be evaluated.
The details of the calculation are quite specific to this particular formalism, in which the string vacuum is specified by fermions on the world-sheet. Also, the object studied is different from a standard model fermion, which needs to be built from massless fermions given a mass by a Higgs-yukawa coupling.
