The majority of cats nowadays are a result of massive amounts of breeding between species over multiple, if not hundreds, of generations (keep in mind that cats can and often give birth when they're around a year old and can breed 3x a year), so 15 generations means around 15 years) to the point where it's all giant mush breed wise.
Unless either they've been super isolated or specific bred as a purebred, then they're categorized less based on breed and more on their physical characteristics.
For instance, a "domestic short hair" is a cat that doesn't have long hair and has been bred within in the area/country.
Tabby cats, like the one in OP's picture, means they have a "M" shaped mark on their forehead. A calico cat means they're a tri-coloured cat. A British long hair cat means they're a cat with long hair that was born in, you guessed it, the UK.
Whenever you see cats in shelters that are labeled as specific breeds Russian Blue for instance, unless they're a purebred, it means they look like one by the roll of the genetic dice.
Unless they're specifically a purebred or specifically bred to be something (in both cases it often have health conditions due to inbreeding), anyone who tells you their cat is a specific breed is either misinformed or trying to deceive you.
There are exceptions, like Bengal cats, but even then for domestication and health purposes they've been bred with domesticated breeds for at least two or three generations minimum.
Tl;Dr: She's a domestic short hair tabby, aka a cat ¯*(ツ)*/¯
(I've worked with different cat rescue organizations and nonprofits, and I'm also fostering two semi-feral cats at the moment.)
I think the more important distinction is whether the cat is bred or rescued. Good on op for making a home for any cat. If at all possible, it's always better not to support breeders.
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