This is Hans's definition:
"Usually what people think of as a classical variation is a short solo dance that elaborates on the character's traits while also displaying the technique of the dancer. Many of them can be modified in various ways to either suit a particular dancer or be interpolated into a different ballet."
And the ballet glossary at ABT contributes this:
Variation. A solo dance in a classic ballet.
But is any solo dance in ballet necessarily a variation? Are there exceptions?
Be that as it may, the term "variation" in music has a somewhat different meaning. In most cases, a set of variations (I can't think of any examples of a single variation) is based on a theme, which may be by the composer of the variations or by another composer - think of Brahms's Variatiions on a Theme by Haydn (which theme is actually not by Haydn at all), or Britten's Young Person's Guide (Variations and Fugue on a Theme by Purcell, better known in ballet circles as the music for Jerome Robbins's Fanfare). In music before the Romantic period, a primary feature of the variations is that they preserve the phrase structure of the theme, often but far from invariably in the same tempo and meter. This is true of the Goldberg Variations, Beethoven's Diabelli Variations, the Mozart set Balanchine used as the last movement in Mozartiana, and many others. One type of variations is known as "doubles," where each succeeding variation uses shorter note values while preserving the underlying tempo; a familiar example is found in the slow movement of Beethoven's Appassionata Sonata. Other types of variations do more to develop motifs from the theme; this is true also of the Goldbergs and the Diabellis, which nevertheless preserve the phrase structure of the theme. But in music of the Romantic period and later, the developmental type of variation comes into play without keeping the underlying phrase structure. And so in the Britten Young Person's Guide / Fanfare / Variations and Fugue on Purcell, the form of each variation is quite free in relation to the theme, but each variation develops one or two main motifs from the theme: an ascending triad, or three notes ascending and then 4-5 descending. I think that once this pattern is recognized, it's easy to identify when hearing the music.
As for the fugue, it's quite common in variation sets to conclude with some section of this type. In the classic variation form, the repetition of the same phrase structure builds a kind of momentum (think of the Mozartiana variations), and a change of pace like a fugue is useful for breaking this momentum and rounding off the composition. Mozart / Tchaikovsky / Balanchine use instead a free cadenza, leading to a slow final variation, and then a final fast coda.
And so the upshot is that I don't think "variation" in the musical and the balletic sense are entirely analogous.