Self-duets of the sort you have in mind are an example of "overdubbing", a technique allowing a musician to record one part and then record a second part while playing back the first. The technique goes back to the very early days of recording technology. Wikipedia has a brief history.
Perhaps the earliest commercial issue of recordings with overdubs was by RCA Victor in the late 1920s, not long after the introduction of electric microphones into the recording studio. Recordings by the late Enrico Caruso still sold well, so RCA took some of his early records made with only piano accompaniment, added a studio orchestra, and reissued the recordings.
This is corroborated on the Victor Records website:
Some highlights of [Nathaniel Shilkret's] career include ... conducting some of the first overdubbed recordings onto Enrico Caruso's voice. (SOURCE)
The earliest example of a "self-duet" mentioned in the Wikipedia article is Sidney Bechet's 1941 recording of "The Sheik of Araby", on which he plays all of the instruments.
However, depending on how expansive you want the definition of "self-duet" to be, keyboardists have been playing self-duets since, at least, the invention of counterpoint for those instruments. One obvious example is the fugue, which requires the keyboardist to play several "voices" simultaneously. For example, here is Bach's Fugue in C Major from The Well-Tempered Clavier Book 1 (BWV 846).
If you require one musician to play to separate instruments, then whoever invented the "one-man band" would be the originator of the self-duet. This web article places one-man bands as early as the 13th century.
Certainly one of the oldest combinations of two instruments played simultaneously is the pipe and tabor, to which the earliest references appear in the 13th century.
Wikipedia corroborates this.
