Can the shortening of proper nouns, such as "The Nuge", be considered a type of elision?

The shortening itself I would term clipping rather than elision.

The use of the definite article with an often altered version of a person’s name (as wonderfully satirized by Rob Schneider as Richard Laymer on SNL, 1990–94) is I think a form of aggrandizement (OED s.v. aggrandize: “4. trans. To cause (a person) [to] appear greater than the reality; to exalt.”)

It is referred to as an agname or agnaming

Styled or called, apart from christian and surname. Urquhart Jewel Wks. 1834, 214 Colonel Alexander Hamilton, agnamed dear Sandy. H. Miller Scenes & Leg. iv. (1857) 47 He was agnamed Gulielmus de monte alto.

Is there a phrase in English that means an apology for banality?

Scientific name for rectangular Pupil?

A word or expression for someone who had a very important influence in your life?

By some counts, only 600 of the 6,000 or ___ languages in the world are “safe” from the threat of extinction

What to call it or someone who easily laughs about something which isn't funny? [closed]

Difference between 'awarding marks' and 'rewarding marks'

What is a noun or adjective to describe somebody who juggles work, study, hobbies, family and more?

Use of [sic] when correcting factual errors in reprinted work

Is there a good word for "unsurpassability"?

Word or expression for someone who intentionally says false statements to bring someone else down?

What's the difference between ‘perhaps’ and ‘maybe’?

Correct use of lie or lay in the following context