It looks like this happens with all fonts. I don't think it's specific to the SVG and PDF backends, since the differences in rendering are visible in the plot window. (I am using the GTK3Agg backend.)

import matplotlib.pyplot as plt
fig = plt.figure()
for (i, fontset) in zip([.1, .3, .5, .7, .9], ['cm', 'dejavusans', 'dejavuserif', 'stix', 'stixsans']):
    plt.rc('mathtext', fontset=fontset)
    fig.text(i, 0.5, r'$\dot s$', ha='center', va='center', size=250, alpha=0.5, color='red')
    fig.text(i, 0.5, r'$\dot{s}$', ha='center', va='center', size=250, alpha=0.5, color='green')
plt.show()

The root of the problem is the difference in the way s and {s} are treated. The former is a symbol (which gets converted into a Char), whereas the latter is a group (which gets inserted into an Hlist).

By default, an Hlist adds kerning to characters, so {s} is actually rendered with some additional space after it. The dot is then centred above the Hlist. That's why it is displaced to the right.

Good catch. The behavior of \dot{s} would seem to be in error, no?

I think that it is an error, yes. At least https://www.ctan.org/pkg/comprehensive writes with {} around the character for accents. (And I always was under the impression that it is the same, but indeed a less authoritative source...)

Indeed a good catch @tfpf !

I agree with oscargus. LaTeX does not add that extra space at the end, so the behaviour of \dot{s} is incorrect. However, LaTeX's behaviour resembles that of \dot{s}, probably because it shifts the accent when the character is italicised. Which might be the purpose of sym.width / 4.0 in the accent function.

The italic correction messes up the placement (which isn't obvious unless the text under the accent is long), while the extra space at the end is responsible for the difference between s and {s}.

I should add that the point about the italic correction has already been noted in #19299.

I think that #23189 actually solves this (and explains one of the failing tests). One can probably make a simpler patch that just handles this, something like:

             if not isinstance(sym, Char):
                sym = sym.children[0]

if it is expected that #23189 will take a long time to get ready (I think that it is probably worthwhile to spend the effort there if we are still planning to "break" how the accents are rendered...).

if not isinstance(sym, Char):
    sym = sym.children[0]

This works well! However, if the group contains another group nested inside it (e.g. \dot{{s}}), the placement of the dot will again be off.

Yeah, you are right. Probably better to do something like:

while isinstance(sym, Hlist):
   sym = sym.children[0]