I think it's the usual meaning of shape, it's just perhaps that we don't typically think about what that really means. [I was an author of the Imperial study mentioned in the article, and I had some long arguments with my collaborators about whether it was right to describe our measurement as the "shape of the electron", so I feel compelled to defend the point!]

If I pick something up on my desk and feel what shape it is, what I'm really doing is mapping out the interaction between the density of electrons in the object under test and those in my fingers. I might infer that the electrons of the object are distributed in a spherically symmetric way (it's round), or perhaps something more complex (all the other less symmetric shapes).

So shape in the context of the electron shape measurements means how it interacts electromagnetically. If it interacted perfectly spherically symmetrically, I think it would be reasonable to say it was round. If it interacted in a more complex way (as in, you could grab it and rotate it, because it has some non-spherically symmetric interactions) then it's not round. Interestingly, the electromagnetic interactions of an electron are extremely tightly constrained by it having only 1/2 unit of spin. You can expand any field around a point in terms of spherical harmonics, and you can show the with spin 1/2, the electron can only interact in the manner of the first two spherical harmonics - monopole and dipole. So it can be round (monopole), or round + a more negative and less negative end (dipole). Nothing more complicated than that. (Assuming you believe quantum mechanics. The Wigner-Eckart theorem is the thing to look up if you're interested.)

These measurements, then are measuring the dipolar component of the electron's electromagnetic interaction. The only way in which it could be not round.

As to the point charge thing: well, that's like your opinion man :-) Which is to say, the electron is what the electron is, and it cares not how humans decide to describe it! These experiments are fine examples of a long tradition of measuring and observing to make sure our theoretical descriptions are actually faithful to reality.

You might be tempted so say that if one of these measurements discovered that the electron was not round, then maybe that would be evidence for the electron being not a point particle. It's complicated though ... and you'd find many physicists would start arguing with you if you did say that, because the current description of the electron - while being a point particle in a certain sense - is already pretty complicated (basically, because of interactions between all of the different quantum fields) and many people would say it's already not point-like. Many wouldn't though. Maybe the point here it's quite tricky to be precise about these things without just doing it properly with maths!

I'll just paste a comment I had on a different thread because it answers your question as well.

You are not to blame for not understanding this, it's just that the analogy for the electric dipole moment coming from a non-spherical 'shape' of the electron is extremely bad. Moreover it's missing the most important reason why we search for EDMs, because the existence of one in an elementary particle would indicate the violation of the time-reversal symmetry (T), which assuming CPT conservation [1] leads to CP violation (Charge conjugation and parity symmetries). CP violation [2] is needed to explain the matter-antimatter asymmetry of the Universe.

A more proper way, in my opinion, to reason about an electric dipole moment is to think in terms of Feynman diagrams. An EDM (or any dipole moment for that matter) is an interaction of the electron with an electromagnetic field, so interaction between an electron and a photon. The most simple such interaction you can imagine is an electron flying in, at one point it absorbs a photon and flies out - that would be the magnetic dipole moment. You can go more complex though - electron flying in, at one point it emits a photon, then the electron interacts with the EM field (absorbing a photon) and then it reabsorbs the photon it has emitted previously. (Note that these analogies are again not perfect as for elementary particles time and space are not the same as in the macro world). Now, it can get even more complicated: If you have an electron it's not really a 100% pure electron. There is always some chance that it transforms for a short time into a quark or neutrino or whatever you can imagine.

When you analyze all such scenarios (electron going into something else, interacting with the EM field and then going back to an electron) some violate CP symmetry, and those contribute to the electric dipole moment. We use that name (dipole moment) as the final result is as if the electron was a ball with some separation between the negative and positive charges and placed into an electric field it experiences some torque. The analogy misses the most important part though, as if it was such a polarized 'ball' it would not violate CP symmetry.

Within the Standard Model the only source of CP violating interactions come from the weak interaction (CKM matrix). These have a very small contribution as the weak interaction is, as the name suggests, very weak. That's why the Standard Model predicts very tiny electric dipole moments. When we are searching for EDMs we are in fact searching for such rare transformations through some new undiscovered particle that violate CP symmetry. If we detect some non-zero EDM that would mean that there is some interaction that is not included in the standard model that violates CP, not that the electron is not a round sphere or a sphere with a bump.

[1] - https://en.wikipedia.org/wiki/CPT_symmetry [2] - https://en.wikipedia.org/wiki/CP_violation