This seems good, but is this really the best place for it within this document? Nothing nearby seems related? Maybe put near annotations?

This was arbitrary, I am definitely happy to move if there are more appropriate places. I wanted to give it a menu entry as this was mentioned in the original issue (#9967)

I would actually go back a step and have the quick-start guide have a section on plotting methods in general that could include patches. Probably with a link to Plot Types to make it short.

I think more detailed commentary belongs in https://matplotlib.org/devdocs/users/explain/artists/index.html.

Do you suggest a new section at that page, or adding a new "Concept: Patches" page? I think I can definitely see the latter - sounds like a nice addition. I'm wondering if we want to have both - a quick mention at the quickstart and a more elaborate document under the page you mentioned? I have no strong feelings either way

I think a quick mention that maybe contextualizes/signposts which artists to look at for what? Something like if we don't have the plot type you need, see Markerstyle for points, LineArtist/LineCollection for lines, patches for regions and AxesImage for images, Contour and Quiver for ... and Artist tutorial to understand all the things?.

If it was just me, I'd add to quick-start a very short section on plot-types, and include patches as a special case of lower level plot types that need add_artist. I'd add a section in user/explain/artists on "Patches". We may want to rename user/explain/artists to make it more clear this refers to plot elements, but not sure what that name would be.

I think this is a reasonable place for this. But not a fan of this almost exhaustive list of Artists, under plot types. Maybe just link the Artists page and have a few examples? eg

Matplotlib plotting methods return a variety of :ref:`Artists <users_artists>` that allow customization of the plot elements.  For example, `~.matplotlib.axes.Axes.plot` returns a :class:`~.lines.Line2D` object and `~.matplotlib.axes.Axes.scatter` returns a :class:`~.collections.LineCollection`.  

and maybe include a short snippet example.

I think this would benefit from a little more context setting: this is what I'm showing you and why-otherwise there's no info here that they're not getting from the comments and code. Also honestly most folks should probably just use plot, axhline, and axhspan to achieve this:

But I like your idea of highlighting the returns objects, and maybe run w/ that the same way we do axes - basically folks almost never call the axes constructor, but modify the object all the time.
So like after

#Plot the data
x = np.array([3, 4, 9, 8, 9, 8, 0, 8, 4, 8])
fig, ax = plt.subplots()

# Computing the mean and standard deviation of the data
mean = np.mean(x)
std = np.std(x)

ln_data, = ax.plot(x, label='Data')
ln_mean = ax.axhline(mean, color='red', label='Mean')
ln_std = ax.axhspan(mean-std, mean+std, alpha=0.1, label=r'$\sigma$')

ax.legend()

give some examples of using the object methods:

ln_data.set_color('orange')
ln_mean.set_linestyle(':')
ln_std.set_hatch('oo')

and maybe end this section w/ a lead in "and for information on creating artists from their constructors, see the artist tutorial'

This makes sense, but what I'm failing to do here is connect this example with the patches - looking at it I don't see patches (of course I know they are used internally, but I would like to expose that somehow to the reader so they understand where they come up). I'm not even sure if this is possible though, so I'm accepting the suggestions and we can maybe think of something later. I think I'm personally also failing to connect the concepts here so I may very well be missing the link :)

Not sure if this is what you mean, but ax.hspan returns a Polygon, which is why ln_std has a set_hatch method, because hatches are 2D texture. Linestyles are 1d texture and here applied to the other line2d object ln_mean.
Which then maybe the call and response form is better:

#line2d
ln_data, = ax.plot(x, label='Data')
ln_data.set_color('orange')

ln_mean = ax.axhline(mean, color='red', label='Mean')
ln_mean.set_linestyle(':')

#polygon
ln_std = ax.axhspan(mean-std, mean+std, alpha=0.1, label=r'$\sigma$')
ln_std.set_hatch('oo')

Little less obvious why you'd do this, but maybe the return object pattern is clearer? Not trying to hold up the PR, but I think like even a comment that axhspan returns polygons (patches) may do the trick?

I think that's what this section is really about, or making custom plot types