Our shower floor is tiled and we don't get any mildew in the sanded
grout lines between the tiles, but we do have an unsightly amount of
mildew in the "silicon-ish" (not sure what the actual matrerial is)
caulking where the tiles hit the walls. I think I already know the
answer, but is there any miracle product that will get rid of that
mildew, or am I doomed to having to remove this mildewed caulking and
replace it with some mildew-resistant variety.

There are a number of mildew sprays, but they basically have bleach in them,
so once you use up the SP14 or other anti mildew spray, pour pure Chlorox in
there, and take advantage of the bleach-resistant pump parts in those
sprayers. Spray once, the surface mildew is gone after soaking for several
hours.
Your caulking likely is one of the cheaper white caulks, which mildew lives
happily on. Best to get a very sharp utility knife blade, razor blade, or
sharp small chisel, and peel it off, cleaning up with a flat razor. Then
clean area spotless, dry completely, before applying silicone bath caulk
from a cartridge, pressing it in with the nozzle as you go. Let set for a
day, in dry conditions.

This is something I'm curious about: Is silicone more mildew resistant
than acrylic caulks? The word I've been getting is that both kinds
of caulk from a given manufacturer contain the same mildew-suppressing
agent. Given that, is there some reason why silicone should nevertheless
be better in this regard?

Yes, I have found GE Silicone bath and tile adhesive caulk is best. Don't
get the paintable, nor the Dap or creamy white products. GE Silicone used
to have small amounts of arsenic to repel mildew. I dont know what the
additive is now, but it works, and the surface of Unpaintable Silicone seems
smoother and shinier, making it easier to remove the mildew once it forms,
and harder for it to get a foothold in the first place...