Fabric question

I have this old quilt my grandmother made that all my kids fight over because of the backing fabric.

I’m currently making them all memory quilts and want to use the same fabric but I cannot find it.

It’s like what newborn outfits were made out of in the 80’s/90’s. Soft, kind of fuzzy. Everyone says terry cloth but all I’m seeing is towel fabric and sweatshirt fabric.

What is that fabric called??