When people talk about patterns that repeat, they usually mean one of the 17 wallpaper groups (https://en.wikipedia.org/wiki/Wallpaper_group). And people are interested in patterns that do tile the grid completely, but don't match one of those patterns. Originally it took thousands, but people had gotten it down to 2 shapes that together tiled the grid completely, but not in one of those 17 groups.

Now, people have gotten it down to a single shape that—by itself—tiles the pattern infinitely, but not in one of those 17 groups.

it’s a non-square shape that does not and cannot repeat

Technically the shape itself repeats like a mug, infinitely tiling the plane. However, that tiling is not overly repetitive -- if it's like the Penrose tiling, it can be self-similar in a handful of rotations about a single origin, but unlike a square tiling, does not admit infinitely many self-similarities.

I mean the obvious answer is to use infinitely large objects, but I think you're just being asinine at this point.

No I really don't understand. I look at the image and clearly see the same shape repeated again and again

You’re right that the shape is used again and again, but the pattern of how it is laid does not repeat. So the new thing here is that there is a single shape that can cover the plane AND that it does this without the pattern repeating (and as an aside it cannot be made to do it with repeats)