This appears to be a myth manufactured by S. N. Sadasivan from oral history. He wrote:
The breast tax was levied as soon as a woman of the Ezhavas and castes below, attained puberty and more attractive the breasts were, the more the tax she had to pay.
No citation is given. The Quora user reports he cannot find a tax where the "more attractive the breasts were, the more the tax she had to pay" mentioned anywhere before Sadasivan's book, and I cannot either. The claim of some versions that women paid for the right to cover their breasts is also incorrect -- they were not permitted to cover whatsoever. However, the head tax on women really was called mulakkaram or "breast tax", and in this answer I will prove that the name of this tax was derived from real beliefs about uncovering breasts.
The story of Nangeli, a woman who supposedly cut off her breasts in 1803 (or "about 1840," according to Sadasivan) to protest the tax, has also been claimed to be a complete myth by some of these Quora users. Local people dispute this and there are multiple oral attestations of the story. I will not attempt to dig up the origins of the story here, but the fact of the matter is that there was a rule that lower-caste people had to expose their breasts, which was enforced by Hindu riots and church burnings.
The actual dispute
I do not reproduce the many petty things the Shanar caste were required to do to respect the high-caste Nair, as these are listed in another answer, but uncovering women's breasts were one of these.
In 1814, the Government of Travancore responded to Christian missionaries requesting permission to allow Shanar women to cover their breasts by ordering that such coverings had to be a partial bodice, in the style of local Muslims; no Shanar women "were ever to be allowed to wear clothes on their bosoms as the Nair women" (Hardgrave 1968:177). Some Shanars attempted to wear a full dress regardless, which caused rioting. The military was called in after several churches were burned, and in 1829, the Government retracted their permission to cover, as follows:
First, as it is not reasonable on the part of the Shanar
women to wear cloth over their breasts, such custom being
prohibited, they are required to abstain in future from
covering the upper part of their body. (Hardgrave 1968:179)
Breast covering remained the central source of anger for upper-caste Nair well into the 19th century.
In the eyes of the Nairs, the missionaries were surely responsible for the spread of the upper-cloth, which had become the symbol of change in southern Travancore. The Nairs’ antagonism against the Nadars’ social innovations and prosperity was only heightened with the anxiety following the abolition of slavery in Travancore in 1855. [...]
With its economic and political base, the frustrations of the
Nair landowners were directed against the pretensions of the Nadar
and their assumption of the breastcloth. “The agitation has been
recently revived,” General Cullen wrote to the Chief Secretary to the
Government, Madras, in January, 1859, “the Soodras asserting that
the Shanar women are constantly assuming the privilege of covering
the upper part of the person, and thereby preventing a recognition
of the caste. . .” Rioting broke out in October 1858, as Sudras
attacked Nadar women in the bazaars, stripping them of their upper
garments. (Hardgrave 1968:180-181)
In conclusion, breast covering was a not a political sideshow but a very central problem in the kingdom of Travancore for roughly a century. The article by Hardgrave (1968) discusses the serious consequences it had which strained relations between the British and the princely state.
Misuse of debunking
There seems to be a use of this real-life history as a political football by both the left and the right in 21st century India. The left has created a seemingly baseless story about breast inspectors, but the right also creates a false debunking narrative.
The Quora posts are one example of this, but another that gets more to the point of this argument can be found in a center-right op-ed in The Hindu published in 2017. This op-ed accepts the Nangeli story but incorrectly disputes its meaning, using debunking to score nationalist and anti-colonialist points. The author first claims that India knew nothing of patriarchy before the arrival of the British:
This was also the land where women enjoyed physical and sexual autonomy, where widowhood was no calamity and one husband could always be replaced with another. The coast was rich with the tales of great women — from Unniarcha of the Northern Ballad of Malabar, an accomplished warrior from Nangeli’s caste, to Umayamma of Attingal, a princess who reigned over kings. These were brave women of towering personality. But in the 19th century, Kerala’s moral conscience grappled not with their achievements as much as the conundrum that their unabashed bare-breastedness presented.
Virtue, as we recognise it today in its patriarchal definition, was not a concept that existed in Kerala. And till our colonial masters — and fellow Indians from patriarchal backgrounds — sat in judgement over the matrilineal streak heavily infused among the dominant groups here, women, their bare torsos, and their sexual freedoms did not in the least attract attention or opprobrium.
In short, the author contends that women were empowered by their bare breasts. He then scolds those who use Nangeli's story to make a point about breasts:
Nangeli too was recast. When Nangeli offered her breasts on a plantain leaf to the Rajah’s men, she demanded not the right to cover her breasts, for she would not have cared about this ‘right’ that meant nothing in her day. Indeed, the mulakkaram had little to do with breasts other than the tenuous connection of nomenclature. It was a poll tax charged from low-caste communities, as well as other minorities. Capitation due from men was the talakkaram — head tax — and to distinguish female payees in a household, their tax was the mulakkaram — breast tax. The tax was not based on the size of the breast or its attractiveness, as Nangeli’s storytellers will claim, but was one standard rate charged from women as a certainly oppressive but very general tax.
When Nangeli stood up, squeezed to the extremes of poverty by a regressive tax system, it was a statement made in great anguish about the injustice of the social order itself. Her call was not to celebrate modesty and honour; it was a siren call against caste and the rotting feudalism that victimised those in its underbelly who could not challenge it. She was a heroine of all who were poor and weak, not the archetype of middle-class womanly honour she has today become. But they could not admit that Nangeli’s sacrifice was an ultimatum to the order, so they remodelled her as a virtuous goddess, one who sought to cover her breasts rather than one who issued a challenge to power. The spirit of her rebellion was buried in favour of its letter, and Nangeli reduced to the sum of her breasts. (Pillai 2017)
We can see from the actual story that even if the mulakkaram was a head-tax, the claim that it "had little to do with breasts" is false. Ensuring that lower caste women uncovered their breasts was a matter of deep concern and anger among higher caste Hindus in Travancore, and this was most assuredly not because they found bare breasts empowering. The term mulakkaram cannot possibly be unrelated to the fact that breasts were visible when the tax-man came to collect, and were unexposed among high-caste women who did not pay the tax.
Furthermore, the lower caste women were not idiots. They understood that uncovering the breasts marked them as powerless and fought incessantly throughout the 19th century for the right to cover. Now, why did they not take more pride in their breasts? Was this the colonial weight of moralizing Victorian missionaries, or was there some kind of male gaze involved in the violently enforced rule that allowed high-caste women to cover their breasts, and refused low-caste women that right while calling the head tax on them a "breast tax"? I think common sense leads us to the latter conclusion. (Academics agree: outcaste "women had
to endure humiliating dress restrictions, which were also ways to mark their bodies as
inferior and sexually promiscuous." Gupta 2014:669)
This op-ed claiming that Nangeli would have been unaware of the concept of modesty is wrong, and although the idea that breasts were taxed by size seems to be a misremembering based on oral history, it is a fact that breasts were forcibly exposed and that the term mulakkaram was connected with this exposure.
Gupta, Charu. 2014. "Intimate Desires: Dalit Women and Religious Conversions in Colonial India." The Journal of Asian Studies 73, pp. 661-687
Hardgrave, R. L. 1968. "The Breast-Cloth Controversy: Caste Consciousness and Social Change in Southern Travancore." Indian Economic & Social History Review, 5(2), 171–187.
Mateer, Samuel. 1871. The land of charity. London : J. Snow and Co.
Pillai, Manu. 2017. "The woman who cut off her breasts." The Hindu, February 17, 2017.