Awwwdorable!
Would love to see the blocks that you have.
From the description, I would read it as sheeting aka bed sheets. We don't see sheeting so much nowadays, so you could just use a sheet or a plain cotton.
Purpose? the way I read the instructions you copied is ..... Applique to the background fabric. Then the interlining fabrics are cut minus the seam allowance and add to the applique block. Do the instructions continue about assembling the quilt sandwich? I am wondering if this is a quilt without batting? and the interlining helps give some "substance" to the quilt?
Personally, I wouldn't fret about the construction of the quilt to match the pattern instructions. Enjoy the blocks you have, and proceed with today's methods and materials, keeping with the heritage sense of it all.
For those interested, here's a link to some pics of the pattern ......
Laura Wheeler Designs Calico Cat Quilt pattern 1583 - Google Search