The Standard Pour Origins
To answer specifically where the idea of pouring 30ml, 60ml, and 90ml began or where it came from is not scientific, but mathematical as a result of business cost & inventory management and, more or less stemmed from the standardizations of fill amounts for wine and liquor, at least when referring to the 30ml amount.
While the 60ml and 90ml amounts are merely multiples of the standard 30ml pout, they also have a different story about them as later explained. But, to answer the question of where or how a standard pour began, you have to start at the point where there was any sort of standardization of alcohol in the first place, which mostly involved only those who produced it on a mass scale.
(It's worth noting that the question asked about 3 specific pour volume amounts and how they allegedly are a standard, but then mentions a measurement specific to India and Nepal as quoted, referred to as the peg & Palatia peg, without any context to their relationship to the 3 mentioned volumes. Therefore it is assumed that the OP infers the peg has a role in those 3 mentioned measurements and the answer below is provided within that assumed context.)
From Standard Bottles to Standard Pours
While historically speaking, on a world trade level, the standardization of the fill amount for wine and liquor was set in Title 27 of the Code of Federal Regulations of 1979, which most countries adopted as part of their desire to participate in world trades as can be seen by the updates being made even as recent as 2020 and witnessed here by the US and here by European nations, suggesting that regulations on trade are what officially set the standard for bottling.
In order to ensure the implementation of the Agreement between the
European Union and Japan for an Economic Partnership (12), it was
necessary to provide for a derogation from the nominal quantities set
out in the Annex to Directive 2007/45/EC of the European Parliament
and of the Council (13) for spirit drinks in order to allow single
distilled shochu produced by pot still and bottled in Japan to be
placed on the Union market in traditional Japanese bottle sizes.
While this might seem to only prove that adjustments will be made in the interest of "Economic Partnerships", it also exemplifies a governing body's desire to regulate the sale and trade of alcohol which lead to the standard bottling and fill size. While in most countries, the most well know standard of bottling size is the 750ml for bottles of wine which has numerous theories of how it came to be depending on who you ask, the point here is that when standard bottling sizes were set and regulated for trade, most countries adopted them for economic trade reasons. And, even India has multiple theories for how the measurement of the peg got it's name according to the Patiala Peg link which has likely since been updated as it now explains:
The Patiala peg is a peg of whisky in which the amount is decided
based on the height between the top of the index and base of the
little fingers when held parallel to one another, against the side of
the standard 26.5 oz (750ml) glass bottle.
The important thing to note is the use of the standard 750ml because it all ties back to that standard 750ml glass bottle:
- When considering a 750ml bottles of liquor, one theory points out that pouring 30ml per glass allows for exactly 25 glasses or "shots" (1 fl. oz. glasses) to be poured and served, an even distribution. So, from global trade, to local businesses like that of a saloon owner, restaurant owner, or drink service business owner, it makes inventory easier to keep track of when you can evenly divide your products for guest sale and consumption.
- When considering a 750ml bottle of wine, one theory suggests that it's because it allows for the standard serving size of 6 glasses of win at 125ml each, used in osterie (wine tavern).
The Need to Measure Any Sort of Pour Amount
Arguably, the only time you would measure out how much you pour is if you were running a business and tracking inventory or if you were trying to ration your alcohol consumption in some way. Otherwise, in summary, the idea comes from a matter of trade and business. AS far as the 750ml standardization, there's plenty of theories out there to argue history, but more or less, it was the most agreeable amount among bottle makers, wine makes, and meal time drinkers.
The question of whether or not the peg originated in India, can be answered by the Wikipedia page, noting the source explaining that liquor companies started making those sizes 60ml and 90ml sized bottles...
...in a bid to encourage more people to try their drinks and attract
In summary, the 30 ml pour originated as a division of the standard fill amount for 750ml bottles, which again, has many different explanations, theories or reasons why it was chosen, while it would seem the pours of 60ml and 90ml are simply a common multiple of 30ml to evenly divide a bottle of alcohol being poured and sold for service as in those amounts, as it allows for a step up in order amount for a bar, tavern or other drink establishment; i.e. a double, a triple while maintaining an even division of the bottled amount.
And, while the pour themselves probably did not originate in India, the peg measurement did and, perhaps because of it's acceptance of the new smaller serving sizes of alcohol companies trying to push more sales there, the country helped make it become a standard there once it was widely accepted.