What is the official name of this game? In India, it is called kaali teeri (3 of Spades)
Regarding the game rules and origin: It appears that this game is not widely know in English on the Internet. However, I found two families of card games played in other parts of Asia that appear to be related.

Regarding the name: "Kaali Teeri" or in English "Three of Spades" is of course a perfectly normal and logical name for a game of this nature. Based on how card game names typically evolve, it may have been coined when the 3 of spades first received its special role - or later in a cultural environment where this is an unusual feature. Some other features of the game may have changed since then, without causing changes to the name. It is quite possible that the game's Hindi name is just a translation from another language. In the same way, it's perfectly sensible and logical to call this game "Three of Spades" in English. Alternatively, one could call it "Kaali Teeri" in English.

The following are all the possible relatives that I could find. Let's start with the Rook family:

Rook is an American card game with special cards, invented in an American Christian community that disapproved of standard playing cards because of their association with gambling. The deck consists of 4 suits with 14 cards each (numbered 1 to 14) plus a joker-like extra card representing a rook bird. The game has many variants, some of which make it more similar to Kaali Teeri. For example, there are variants in which the player can call a partner instead of playing in fixed partnerships. Remaining major differences:


The game is only played with ranks 5 to 14, corresponding to the ranks 5 to King and Ace (high) when playing with ordinary cards, plus the Rook, corresponding to a Joker or perhaps the 3 of Spades.
Card values are 5: 5, 10: 10, Jack/Queen/King: 0 (not 10), Ace: 10, Rook: 20 (not 30).
Every player gets only 9 of the 41 cards. The remaining 5 cards form a 'nest', from which the highest bidder may improve his hand.


Shelem or Rok is an Iranian game that is obviously connected to Rook. Major differences to Kaali Teeri:


There is no equivalent to the Rook, Joker or 3 of Spades.
Card values are 5: 5, 10: 10, Jack/Queen/King: 0 (not 10), Ace: 10.
Every player gets only 12 of the 52 cards. The remaining 4 cards form what is normally called a widow in English, from which the highest bidder may improve his hand.


Another apparent Rook variant is Oochlee, played by the Christian minority in Syria.

Now we come to the picture-taking family:

Etori is an almost extinct Japanese game. A modern variant that is still played is called Kakeya Toranpu. Major differences to Kaali Teeri:


The partnerships are fixed.
The trump suit is not determined by bidding but in one of several much less interesting ways.
Regardless of trump suit, the highest card is always the Ace of Spades (not the 3 of Spades). Optionally a Joker may be added which is then the highest card. (It is not reported what other card if any is removed in this case.)
The object is for a partnership to win more than 8 of the 16 scoring 'picture cards' Ace, King, Queen, Jack of all four suits. (This is equivalent to making Ace, King, Queen, Jack score 10 and everything else 0.) For this reason, this kind of game is called a picture-taking game.


Japanese Napoleon has a lot of differences to Kaali Teeri, but it is obviously related to Etori and has some features that are actually closer to Kaali Teeri than Etori is:


10 is also a counting card, making the scoring system even more similar to Kaali Teeri.
Players bid for the game. The winner calls for a partner by calling for a card.
There is a variant in which a Joker is added as a 'trump hunter' and the 3 of Spades(!) has the role of a 'Joker hunter'.


The Korean game Mighty is very similar to the Japanese games described before.

Overall, I would bet that Kaali Teeri connects the American game Rook and its close relative in Iran with the Japanese picture-taking games. This even fits the geographic situation very well.