I see the same thing in both Firefox and Safari, on a non-P3 display: the original image is solid red, but the copied image shows the logo. So it looks like a conversion did actually happen at some point.
The transform to display the "solid red" block is actually the post-transformed data. The clipboard (rightfully) is carrying the original source buffer.
The issue would be the mechanics of the copy (attaching metadata) and pasting (interpreting the metadata in conjunction with the source image buffer.)
So technically, no, there's no conversion on the source data, it's the representation in the pasted context that's the issue.
The clipboard here is probably copying the image buffer in the untransformed state, and pasting it into MSPaint would dump those values incorrectly.