How did 'equity' semantically broaden to mean 'common shares'?

I ask about its meaning for stocks here, not Equity = Assets — Liabilities. See Personal Finance For Canadians For Dummies (2018), p 217.

Equity — not to be confused with equity in real estate — is another word for stocks.

Etymonline:

early 14c., "quality of being equal or fair, impartiality in dealing with others," from Old French equite (13c.),
from Latin aequitatem (nominative aequitas) "equality, uniformity, conformity, symmetry; fairness, equal rights; kindness, moderation,"
from aequus "even, just, equal" (see equal (adj.)).
As the name of a system of law, 1590s, from Roman naturalis aequitas, the general principles of justice which corrected or supplemented the legal codes.

  1. What semantic notions underlie the nouns bolded overhead with definition 2 beneath?

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The OED indicates that the relevant evolution of senses was

1. The quality of being equal or fair [1315]

3. The recourse to general principles of justice to correct or supplement the provisions of the law [1574]

4. In England (hence in Ireland and the United States), the distinctive name of a system of law existing side by side with the common and statute law. [1591]

5. a. An equitable right, i.e. one recognizable by a court of equity. [1629]

5. c. The ordinary shares of a limited company. They carry the right to the residue of a company's assets after it has paid all its creditors, and share in the distribution of profits. [1904]

In other words, an equity [share] carries an equity [a legal entitlement] in Equity [the legal system], which applies equity [the principle] in order to achieve equity [fairness and justice].