It's not a "Pakistani" heritage. The rice was traditionally cultivated along Indo-gangetic plains of which a small section unfortunately lay in Pakistan.
Historically India faught patent battles for Basmati rice and other traditional knowledges and cultivaars while you watched from the sidelines. So indeed this is a loss. But on the better side such contested GI tags prevents expansion of Basmati rice farming.
Taking liberties as usual....
Basmati rice is cultivated across majority of Pakistan's Punjab plains and northern Sindh.
In the past Pakistan & India jointly fought to keep other from using the name....
Where is basmati traditionally cultivated?
Basmati is associated with the
foothills of the Himalayas and the Indo-Gangetic plain, particularly the Punjab region. Because Punjab predates the 1947 partition as a single cultural/agricultural region, the traditional basmati-growing area ended up on
both sides of the India–Pakistan border.
In Pakistan, the traditional area is principally Punjab, especially the
Kalar tract between the Ravi and Chenab rivers, historically one of the most important basmati-producing areas. (
The Indian Express)
In India, traditional basmati cultivation is concentrated in:
- Punjab
- Haryana
- western Uttar Pradesh
- Uttarakhand
- Himachal Pradesh
- Delhi region
The currently defined Indian GI area is narrower and covers parts of seven states/regions. (
The Indian Express)
So it would be misleading to describe basmati simply as an exclusively "Indian" or exclusively "Pakistani" crop.
Its traditional agricultural geography predates Partition and spans the Punjab/Indo-Gangetic region now divided between the two countries.
And yes—the two countries actually cooperated over the name in the late 1990s
This is the particularly interesting part of the history.
The famous episode wasn't initially
India vs. Pakistan. It was essentially
India + Pakistan vs. an American company.
In
1997, the US company
RiceTec obtained patents concerning new rice varieties derived from basmati strains. The dispute became particularly controversial because RiceTec marketed varieties using names such as
Texmati and
Kasmati, and there was concern that American-grown rice could appropriate the commercial reputation of basmati. (
The Indian Express)
Then in
1998, India and Pakistan
jointly petitioned US authorities to prevent American-grown rice from being marketed simply as "basmati." The WTO's historical account explicitly describes the two countries as jointly filing the petition. (
World Trade Organization)
That cooperation is significant because it recognized a basic reality:
Basmati was a shared geographical and agricultural heritage of the region, rather than something that could straightforwardly be claimed by one post-1947 state.
The eventual US legal outcome was somewhat complicated. The US authorities considered "basmati" a generic term in some contexts, while the US Patent Office subsequently restricted RiceTec's claims and prevented the company from simply monopolizing the name. (
World Trade Organization)
But the really interesting twist came later
After jointly defending basmati against the Americans,
India and Pakistan eventually began competing with each other over geographical-indication protection.
India established domestic GI protection for basmati in
2016. Pakistan subsequently sought its own GI protection, and India applied to the EU in
2018 for protection that would effectively give Indian basmati exclusive GI status in the EU. Pakistan objected because it is also a major basmati producer and exporter. (
The Indian Express)
In fact, in
2006, the two countries were still discussing a
joint international intellectual-property regime for basmati. Indian and Pakistani rice-export organizations and governments were considering jointly registering basmati as a GI/common trademark. (
The Economic Times)
So the historical sequence is roughly:
Pre-1947:
Punjab/Indo-Gangetic region → traditional basmati cultivation on what is now both sides of the border
1997–2001:

India +

Pakistan
cooperate against RiceTec/US appropriation of "basmati"
2006:


Both countries discuss
joint international protection of basmati
2010s onward:
The relationship changes → India seeks its own GI protection
2018 onward:

India seeks
EU GI protection for Indian basmati →

Pakistan objects
2023 onward:
Pakistan files its own EU GI application, creating the modern India–Pakistan basmati dispute. (
Financial Times)
So
your recollection is basically correct, although I'd phrase it as:
India and Pakistan jointly fought to prevent a US company from appropriating/monopolizing the basmati name in the late 1990s, rather than saying they were fighting to make the name "exclusive" to both countries. Their position was essentially that
basmati was a traditional product of the South Asian/Punjab region and shouldn't be appropriated by a foreign producer.
And there is a fascinating historical argument behind Pakistan's claim:
pre-Partition Punjab's agricultural and cultural history doesn't map neatly onto today's international borders. That's why the later attempt by either country to claim exclusive ownership of the name becomes considerably more complicated. (
or.niscpr.res.in)