See this screenshot from SpaceEngine:
Many other examples from websites etc. Do they have a name? I couldn't find anything. If I'm not mistaken, those should be the z-coordinate of galactic coordinate system.
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1$\begingroup$ Not a complete answer and there may not be consensus but Horizons calls them "Drop lines" in the orbit viewer; XEphem calls them "legs" (These are in the context of solar system objects with the lines above/below the ecliptic; not sure I'm seen it for stars above/below the galactic equator outside of video games before...) $\endgroup$– astrosnapperCommented Jul 6, 2022 at 20:16
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1$\begingroup$ I belive they're called stems, based on various google searches pointed me at 3d Stem Plots $\endgroup$– notovnyCommented Jul 6, 2022 at 20:28
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$\begingroup$ Interesting question! What follows is speculation: yes assuming that what's shown are positions in galactic coordinates -- which could be expressed equivalently in either cartesian, cylindrical or spherical forms -- what is illustrated here looks like the cylindrical coordinate form, and $\mathbf{\hat{z}}$ is most likely going to be defined the same way in cartesian and cylindrical coordinates. But the term for the draw lines that simply serve to guide the eye might be "lines of projection" connecting the 3D position with its projection on the $xy$ or $r \theta$ plane. $\endgroup$– uhohCommented Jul 6, 2022 at 20:53
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2$\begingroup$ Is it the ecliptic? Or the galactic plane? There's quite a difference! astronomy.stackexchange.com/q/28071/16685 $\endgroup$– PM 2RingCommented Jul 7, 2022 at 5:15
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1$\begingroup$ This chart library just calls them "bars". sthda.com/english/wiki/… $\endgroup$– Greg MillerCommented Jul 9, 2022 at 15:08
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Apparently they are called differently in different places. Drop lines, legs or stems.