There is no way to reference a specific quote on Wikiquote.

Why and when would anyone want to "reference a specific quote on Wikiquote"? Use case welcome.
You can provide a permanent link to any revision of a page or a section. If you want to reference (in a scientific sense) a specific quote then you'd need to reference the original source of that quote, and the original source is never Wikiquote.

If you need to reference anything, it should be the primary source, not WikiQuote

In T187900#3989653, @Aklapper wrote:

Why and when would anyone want to "reference a specific quote on Wikiquote"? Use case welcome.

Let's say I want to share this quote with someone:

I am a slow walker, but I never walk back.

I could... copy the whole quote, which loses where it came from, and also the citation of where it originally appeared. Or I could just send you a URL to the quote and you will get the source (wikiquote) the full text of the quote as well as the full citation.

This is exactly like how on Phabricator you can link to a specific comment, or on Facebook you can link to a specific post, or on Twitter you can link to a specific tweet.

I honestly do not see the utility of Wikiquote without this feature.

In T187900#3989663, @Weslima wrote:

If you need to reference anything, it should be the primary source, not WikiQuote

Of course, but that doesn't stop people from linking to Wikipedia pages.

In T187900#3989680, @dbarratt wrote:

This is exactly like how on Phabricator you can link to a specific comment, or on Facebook you can link to a specific post, or on Twitter you can link to a specific tweet.

That's what sections are about. So you seem to ask Wikiquote to make every <li> element a section.
I don't know of a website that provides IDs to link to the x'th item in a list by default...

In T187900#3989788, @Aklapper wrote:

That's what sections are about. So you seem to ask Wikiquote to make every <li> element a section.
I don't know of a website that provides IDs to link to the x'th item in a list by default...

I'm saying the content model might need to change.

Wikiquote is a free online compendium of sourced quotations from notable people and creative works in every language, translations of non-English quotes, and links to Wikipedia for further information.

https://en.wikiquote.org/wiki/Main_Page

If the primary content is "sourced quotations" then that's what we are storing / retrieving. Each sourced quotation should have an id and be indexed and linkable.

Right now, the content model is:
Notable Person / Work → Section → Sourced Quotation
and yeah, in that model, why would something two levels down have an id?

But, as per the description of the site, the content model should be:
Sourced Quotation → Notable Person / Work
In this way, a Notable Person / Work is more of an "author" than an object.

A way to express this is kind of like twitter:

Here's a link to all of Obama's (the author's) quotes;
https://twitter.com/BarackObama
and here's a link to a specific quote:
https://twitter.com/BarackObama/status/964185606803853314

Wikiquote, I think, should work more like this, and less like Wikipedia.

The purple-numbers idea is about adding anchors to every paragraph or list-item (etc.) as little purple-coloured hashes or pilcrows at the end of the block. I don't suppose it would ever be considered on a Wikimedia wiki as it's often considered a bit annoying to readers, and anyway seems to have died out in recent years.

Some relevant links:

• https://en.wikipedia.org/wiki/Purple_Numbers
• https://www.dokuwiki.org/plugin:purplenumbers
• http://scripting.com/2014/02/23/designChallengeParagraphlevelPermalinks.html
• http://www.tbray.org/ongoing/When/200x/2004/05/31/PurpleAgain
• https://www.mediawiki.org/wiki/Extension:Purple_MediaWiki