Does voice doubling an instrument's melody while improvising have a name?

The answer is no; the style of scat singing in unison along with an instrumental solo, or doubling it at an octave (like George Benson) does not have a specific, concise name (as far as I can tell).

In the case where an instrumentalist doubles their own improvisation with voice, there is not a specific term. Slam Stewart made his name doubling his bass in this way, having gotten the idea from Ray Perry, who did the same in his violin solos (recording not readily available). In general, it's considered something of a novelty.

Some musicians famously vocalize while they play, but not with the intention of, say, Slam Stewart. Standout examples are Glenn Gould and Keith Jarrett. In this video, Glenn Gould is actually singing along (about 5 sec. in), though not with the intention of featuring his voice. And here is a compilation of Keith Jarrett vocalizations.

Vocalese is the term for adding lyrics to and singing an instrumental melody. The term is sometimes also applied to any vocal realization on an instrumental part.

Wikipedia has a good entry on the term and some history of the practice.

To pick just one particularly well-known example, Eddie Jefferson turned James Moody's improvisation on "I'm in the Mood for Love" (link to Frances Langford's premier of the song) into a vocalese version.

Here is a YouTube link to James Moody's version. (Introduction by Dizzy Gillespie; James Moody begins at 1:45) And here is "Moody's Mood for Love" by Eddie Jefferson.