In this image the American astronauts wear some patches on their chest but what is it?
If it is the expedition's emblem, why it is so different from Russian ones?
I am pretty sure that is the Expedition 66 patch (link). Note the two yellow diagonals, which match to the tops of the gold 6's.
The Soyuz crew are wearing what seems to be a specific patch for Soyuz MS-21. It's not very visible in the photo here, but this pre-flight photo (archive.org as the original seems to be unavailable) shows a patch on the shoulders of their launch suits, and it's a bit easier to see the details. It seems to match up to the ones in this photo.
Finally, the one on the chest of the man on the left (Korsakov?) seems to be the Expedition 67 patch, which is also visible on the pre-flight photos.