This is not a thorough study on measuring cups during the 19th century. However there are a few tidbits to be aware of.
For example in the 1810 The New Family Receipt Book they don't have a measurement of one cup. They refer to a common tea cup, ordinary tea cup, one coffee cup.
I have found images of (Victorian) pewter measuring cups on the Internet. I've also found tin measuring cups dating back around 1840.
The earliest cookbook that I found any measurements for was in the 1830 Seventy-five Receipts for Pastry Cakes and the measurements were in pounds, ounces and quarts, pints. However there do have "A tea-spoonful of salt." The only cup measurement was again a tea-cup. 1/2 pint I started to referenced around this time as well, but never referred to as one cup.
By the end of the century I found glass measuring cups available as well as some high end copper and brass measuring cups.