I think what you are looking for is a solo piano instrumental nostalgia standards ballad, which isn't necessarily a recognized genre, but it is a findable type of music. Basically, "nostalgia" standards are lightly jazz-influenced classic popular American songs of the 20s through 50s --also known as the "Great American Songbook".
As Time Goes By
There are also some newer songs in this same style:
City of Stars
Original artists in this style range from Nat King Cole to Cole Porter, Fats Waller to Frank Sinatra (and yes, Duke Ellington). Recorded versions of this music would have typically been with a big band; but if you were an amateur pianist of the era --like your grandfather--, this would have been the style you would have naturally gravitated to. A lot of music of this type is more up-tempo and jazzy, but there were a fair number of ballads as well.
Cole Porter (cover) - Piano Instrumental
Nat King Cole - Solo Piano
The specific flavor of your grandfather's playing may have well been particularly influenced by Italian-American composers and performers such as Sinatra, or songwriter Henry Mancini.
You might also have luck considering this as a piano version of the lounge music genre. Lounge music was relaxing background music played in bars and "lounges" in the 50s and 60s. It was often solo piano, with or without vocals, and included a lot of nostalgia standards, as well as some other "relaxing" styles (the modern version is called "chillout").
Piano Lounge Music