A little while ago I was looking through a book I got from the library: "Little Quilts All Through the House" by Alice Berg, Mary Ellen Von Holt, and Sylvia Johnson, printed in 1993. I want to make some doll quilts for my granddaughters and this book has a lot of country style, cute, easy quilts, both pieced and appliqued. When I turned to the Applique section, lo and behold, the first stitch described and pictured is very similar to the stitch your MIL used. It's just called the applique stitch and the only difference is that the stitches are shown closer together than on your MIL's quilt top. The section goes on to include needle turning and buttonhole and blanket stitches. I saw your post earlier, so thought I'd share this.
I'm pretty sure I've seen this same stitch used on other appliqued quilts, although I don't remember where. I'm sure they were probably either country or folk style quilts.
I'd love to see pics of the quilt when you finish it.