SM is a pragmatic model. It says that elementary particles have zero size but it allows for an effective size known as the cross-section which is basically the effective area two elementary particles can occupy before they scatter from each other at large angles. All elementary particles have non-zero cross-sections. The non-zero cross-section indicates that there is an effective core where the hard-scattering takes place but the spatial extension of the particle is theoretically zero.
String/M theory rejects the assumption that elementary particles are point-like with no spatial extension. According String/M theory the elementary particles have a tiny (infinitesimally small) size.
From a classical point of view, if elementary particles have zero size then they cannot have a non-zero electric dipole moment. If they have a spatial extension as the String/M theory claims then it is possible for them to have a non-zero electric dipole moment. For the definition of classical electric dipole moment you can look here.
SM avoids this kind of talk at all cost. It is not just the SM, its parents the Quantum Field Theory (QFT) and Quantum Mechanics (QM) also avoid this kind of talk. Instead, these theories say that electron electric dipole moment (EDM) is an intrinsic property just like the electron spin. There is no classical counterpart of the electron spin. Similarly, there is no classical counterpart of the EDM.
Officially speaking, EDM is zero in SM but as I said earlier, SM is a pragmatic theory, it allows for a tiny non-zero EDM through various extensions. By measuring the EDM precisely we can determine which extensions of the SM are better candidates to describe physics beyond SM.
The experimental measurements by the ACME collaboration established a new upper limit 12 times smaller then the previous upper-limit. The new upper limit is
The smaller the upper limit the less chance of finding new physics beyond SM.
According to the Resonaances blog the simplest explanation of the current data is that there are no supersymmetric particles up to at least ~10 TeV. This is beyond the reach of LHC at CERN. This is very bad news for the theories employing supersymmetry.
The ACME collaboration claims that they will improve the precision of their measurements by another order of magnitude soon. That may be bad news for some of the BSM (beyond SM) theories.
Update: ACME II collaboration has a new upper limit for electron EDM. Please see my blogpost about this.