Answer:
What you describe in your text is known as a swarm plot (or beeswarm plot) and there are python implementations of these (esp see seaborn), but also, eg, in R. That is, these plots allow adjustment of the y-position of each data point so they don't overlap, but otherwise are closely packed.
Seaborn swarm plot:
Discussion:
But the plots that you show aren't standard swarm plots (which almost always have the weird looking "arms"), but instead seem to be driven by some type of physics engine which allows for motion along x as well as y, which produces the well packed structures you see in the plots (eg, like a water drop on a spiders web).
That is, in the plot above, by imagining moving points only along the vertical axis so that it packs better, you can see that, for the most part, you can't really do it. (Honestly, maybe the data shown could be packed a bit better, but not dramatically so -- eg, the first arm from the left couldn't be improved, and if any of them could, it's only by moving one or two points inward). Instead, to get the plot like you show, you'll need some motion in x, like would be given by some type of physics engine, which hopefully is holding x close to its original value, but also allows for some variation. But that's a trade-off that needs to be decided on a data level, not a programming level.
For example, here's a plotting library, RAWGraphs, which produces a compact beeswarm plot like the Politico graphs in the question:
But critically, they give the warning:
"It’s important to keep in mind that a Beeswarm plot uses forces to avoid collision between the single elements of the visual model. While this helps to see all the circles in the visualization, it also creates some cases where circles are not placed in the exact position they should be on the linear scale of the X Axis."
Or, similarly, in notes from this this D3 package: "Other implementations use force layout, but the force layout simulation naturally tries to reach its equilibrium by pushing data points along both axes, which can be disruptive to the ordering of the data." And here's a nice demo based on D3 force layout where sliders adjust the relative forces pulling the points to their correct values.
Therefore, this plot is a compromise between a swarm plot and a violin plot (which shows a smoothed average for the distribution envelope), but both of those plots give an honest representation of the data, and in these plots, these closely packed plots representation comes at a cost of a misrepresentation of the x-position of the individual data points. Their advantage seems to be that you can color and click on the individual points (where, if you wanted you could give the actual x-data, although that's not done in the linked plots).
Seaborn violin plot:
Personally, I'm really hesitant to misrepresent the data in some unknown way (that's the outcome of a physics engine calculation but not obvious to the reader). Maybe a better compromise would be a violin filled with non-circular patches, or something like a Raincloud plot.
