Also Mandelbox (negative scales, here is -1.5):
 
and

from http://web.archive.org/web/20151229163454/http://www.miqel.com/fractals_math_patterns/mandelbox_3d_fractal.html
and https://digitalfreepen.com/mandelbox370/

It can show up with some logical operations (XOR?). For example,

http://spanishplus.tripod.com/maths/FractalSierpinski.htm

It can show up with some logical operations (XOR?). For example,

http://spanishplus.tripod.com/maths/FractalSierpinski.htm

This one really surprised me! The AND method is remarkably simple. The code just checks if X and Y values equal Zero when ANDed together and color accordingly. The rest is just a quirk of math, I suppose! I'm curious if similar results would work for non-binary equivalents of AND...

I'm sure I've come across Sierpinski Triangles when I wasn't expecting it... usually when I wanted some more unique results and therefore I'd get annoyed about it and not really pay attention to how it got there. But one example I do remember, which might not be exactly what you're looking for, is when I was making a collapsing structure in Minecraft out of sand and torches.
I'd start with a single torch, and placed sand only on top of torches, then placed torches only on the sides of sand where there was only one sand block to chose as the attachment point in that space. (It seemed inelegant to chose one side or the other, so I'd chose neither) So if two sand blocks were separated by only one air, no torch would go there. But if there were three blocks of air between, both sands would get torches on their sides. Soon the familiar triangle took shape.

The fun thing about it is that when you break the bottom torch, the whole thing collapses.

For whatever reason the S-triangle likes to pop up in some sea shells! See the picture attached

Not sure if this counts but there's the Eiffel tower.

If not, then there's trusses (roof constructions) based on the sierpinski triangle:
https://www.researchgate.net/publication/271197420_Fractal-Based_Generative_Design_of_Structural_Trusses_Using_Iterated_Function_System/figures?lo=1&utm_source=google&utm_medium=organic