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!