I believe the term "seam" came from Michael Feathers' "Working Effectively with Legacy Code": “a seam is a place where you can alter behavior in your program without editing in that place”.
I've associated it with the concept of "carving nature at the seams", except that I just realized that in the standard phrasing it's actually "at the joints", not "seams".
I read that book right at the beginning of my career over 20 years ago, when I was the new guy who had actually studied software engineering among all my entirely self-taught older colleagues. Lots of big messes to clean up in that codebase!
https://martinfowler.com/bliki/LegacySeam.html
Claude is using it a bit liberally, but not totally incorrectly.