It is very traditional to quilt 1/4-inch in from the seam line. That, plus the even spacing of the stitches, plus the pin basting, plus the fact that you found the quilt in a hoop, indicate that the stitches you found are her quilting stitches. Hand quilting is often done in a hoop, whereas basting is never done in a hoop.
Are the pins safety pins or straight pins? Typically safety pins are used. Extra pins are needed around the edges to keep the layers secure.
Quilting stitches are counted both top and bottom so, if you are seeing 4 stitches on top per inch, then that would be counted as 8 stitches per inch -- pretty standard for beginning quilters and very acceptable for a utility quilt or charity quilt. Experienced hand quilters strive for 12 stitches per inch. Most likely she was planning to quilt inside every square; however, the most efficient way to do this is to quilt in straight lines -- first one way, then another.
Are the quilting stitches just a single thread? It sounds as if your grandmother had read up on quilting but hadn't quite figured out how to bury her quilting knot and thread tail. Making small quilting stitches takes a lot of practice. Since you don't have other quilts from your grandmother, my guess is that she was just starting out to hand quilt and that's why the quilting stitches are rather large.
The needle looks relatively large in the photo. That would also be typical of a beginning hand quilter.