Wikipedia explains that Delmarva is a term that originated in the 20th century. What was the name of the Delmarva Peninsula during the British colonial time and on 19th-century United States?
Originally it was called "Virginia." :)
In all seriousness, the Chesapeake Bay was the original focal point of the colony of Virginia, moreso than the land. As such, the name "Eastern Shore" (still the term I and other locals use) was the more common term. Even places like Assateague and Chincoteague - completely on the Atlantic and thus having water to their East, are still referred to colloquially as the Eastern Shore.
When the Maryland colony took hold in the later 1600s, the "Eastern Shore" still applied to both colonies.
Delaware as a colony is only on the Atlantic, and so usually if you were talking about Delaware, you identify it as such.
Note the term "Western Shore" is not used. Instead, we speak of "Tidewater" Virginia or up north, just plain "Maryland." Maryland exists on both sides of the bay, but there is no term of which I am aware to distinguish "the mainland" from the Eastern Shore.
Historically, Tidewater was always the economic powerhouse. The first settlements (Jamestown, Williamsburg, etc.. on "the Peninsula" or even "the Virginia Peninsula") were the things that mattered. The farms on the Eastern Shore were always much further removed and isolated economically. Indeed, until the Chesapeake Bay Bridge opened in the 1930s, there were only a few ferries that could even get you there. The British never bothered trying to occupy it during the Revolution, and as an entity, people just didn't refer to it much period.
Like Tidewater, there just isn't a lot of Maryland on the Eastern Shore either - its got a fair amount of poultry, but to a lot of people its just "that stuff you drive through to get to Ocean City," on the Atlantic coast.