What is the origin and extent of the Indian English usage of "only" to emphasize something?

To judge from the comments, it's a calque of the word ही (hee) in Hindi and the other northern languages and of தானே (tāṉē) in Tamil and the other southern languages.

It's easy to see how ही (meaning "nothing else") was translated into only (meaning "nothing more"), since only is 25 times more common than a more precise translation like exactly. Given that there are a billion speakers of Indian English, that translation is arguably now correct.

To add to the theories in OP and the answer of @Malvolio, there's another interesting clue here.

As @NeilW noted in the comments, the British actually used to use the word "only" after the amounts on cheques, receipts, bills, etc. (Apparently to prevent tampering, thanks @MattBishop .) This is still practiced in India today. For example, here's one of my receipts:

So, as @NeilW pointed out, this usage of "only" in relation to monetary amounts is something India inherited from the British, since it used to also be practiced by the British themselves.

I tend to agree with @Malvolio that it's probably a calque. And this monetary related usage may explain why Indians ended up using the word "only" as a replacement for the Indic and Dravidian emphasizers.

During colonial times (and even after), India's national banking system and financial structures were based on a British model and the English language.

Only would have been a word many people were exposed to every day. After all, everyone, no matter who they were, had to deal with money. Paychecks, receipts from shops, cheques, "money orders", price tags, advertisements. The word only would have been (and still is) ubiquitous in society, in relation to monetary amounts.

Combine this with the strong habit from Indic and Dravidian languages to use emphasizers at the end of sentences. It's easy to understand that Indians felt a need for an equivalent word in English. And as I explained, many would have been exposed to the word only every day, in relation to money. Thus, the word could have easily slipped into usage in other contexts from there.

According to the Indian user @NVZ, the phenomena described in OP is common all over India.

The word "only" is often used by many Indians as a replacement for a certain kind of emphasis that is found in a lot of Indian languages. Let's say you are describing somebody who's a habitual liar. A lot of Indians might say, "So, he lied to you ? Of course, he is like that only ". Indian languages usually have suffixes or separate words to emphasize this typical behavior of the subject in question.

In correct English, this would be something like, "So, he lied to you ? Of course, he's just like that". There's a stress on "like" which makes this sentence different from "Of course, he's just like that".

This can be a difficult thing for most Indian speakers to grasp, because firstly, in most Indian languages, stress is communicated through either a prefix, suffix or a separate article-like word. The idea that the same sentence can mean different things when spoken differently, doesn't exist in languages derived from Sanskrit.

Secondly, we learn English almost entirely through written material. Native English speakers learn the language mostly by hearing and pick up the stresses while learning the language. We usually don't get that chance and so, apply a logic similar to Indian languages in situations where stress is important in expressing an idea.

Lets research a bit deeper

"Only" can be a conjunction, an adverb or an adjective.

It is not correct to end your sentence with a conjunction, but nobody would ever do that anyway.

It is fine to end your sentence with an adjective, but "only" as an adjective generally occurs in complete expressions such as "only child" or "the only one" in which it is systematically followed by a noun. In other cases, it is generally replaced by "alone" or "lonely".

The only remaining case is "only" as an adverb. Here it is perfectly legitimate to end a sentence with "only", although it is often poor style.

The usage is explained here.

There are only two cases where this stylistic aspect can be ignored. Emphasis - Sometimes, only is used at the end of a sentence to emphasize the exclusivity referred to by it, i.e. to emphasize that the word only is of prime importance. (This bar is for members only.) To prevent ambiguity - Sometimes, ending the sentence with only is the only way to prevent ambiguity, precisely because of the emphasis mentioned above. (Take one cookie only.) Note that in this case, only qualifies the phrase one cookie. (Take only one cookie, on the other hand might have only qualifying either one or one cookie.)