Why are phonewords called vanity numbers?

A vanity number is a local or toll-free telephone number for which a subscriber requests an easily remembered sequence of numbers for marketing purposes.

While many of these are phonewords (such as 1-800-Flowers, 313-DETROIT, 1-800-Taxicab or 1-800-Battery), occasionally all-numeric vanity phone numbers are used.

Accorging to Google Books the expression is from the late ’80s but why ‘vanity’?

Is vanity number the more common expression to refer to these often used numeric/alphanumeric numbers?

Is it only AmE usage or is it used also in British or other English dialects?

The AHD's primary meaning of the noun vanity is typical:

Excessive pride in one's appearance or accomplishments; conceit.

It follows that a vanity product or service is one that is purchased in order to feed one's vanity, by drawing attention or by artificially boosting one's stature to outward appearances. The oldest derived term in the OED is vanity publisher, attested from at least 1922, a publisher who publishes only at the author's expense. In other words, a vanity publisher is a press that someone pays so that he or she can call him- or herself a published author, in contrast to a traditional publisher, which would pay the author for printing rights, as well as in contrast to self-publishing, where the author produces and maintains control of the entire product.

A more familiar term is vanity plate, for a car license plate on which the registrant chooses a custom/personal word or slogan instead of having random letters or numbers assigned.

Vanity phone numbers, vanity URLs, vanity DNS nameservers… these are all very mildly disparaging terms for ways where someone can pay a little extra money to stand out from the crowd.

Usually things are given the "Vanity" prefix when you're paying extra to get something that doesn't serve a functional purpose.

As @tmgr noted, the most common analogy is Vanity Plates, a car license plate that spells a word, but doesn't add anything, Like this one for £4000. Vanity publishing, is where an author pays for something to be published instead of the publisher paying the author, which is the more traditional situation.

The underlying premise is that the person buying this vanity service wants it for nothing more than to show off. It is potentially slightly misplaced with respect to these phone numbers, as they serve the purpose of making the businesses' number more memorable than a rival service. Very useful in the pre-mobile era.

From Google dictionary

vanity 1. excessive pride in or admiration of one's own appearance or achievements. "it flattered his vanity to think I was in love with him" synonyms: conceit, conceitedness, self-conceit, narcissism, self-love, self-admiration, self-regard, self-absorption, self-obsession, self-centredness, egotism, egoism, egocentrism, egomania;