Based on an CNBC piece with similar claims, the Guardian letter seems to be talking about cobalt:
For example, China controls 80 percent of the market in chemical cobalt, a crucial ingredient in lithium-ion batteries, [Benchmark Mineral Intelligence analyst Caspar Rawles] he said.
Oddly enough a techcruch piece is somewhat contradictory
Approximately 97 percent of the world’s supply of cobalt comes as a by-product of nickel or copper (mostly out of Africa).
Digging the Benchmark report that all seem to using for this data, it clarifies that the Chinese control is by their acquisition of African mines, in DRC in particular:
In 2015, the last full year of production,
102,000 tonnes of cobalt (all forms) of
which 59% was mined from the Democratic
Republic of Congo (DRC) as a by-product of
Outside of the DRC, there are no
significant production hubs with the second
largest producer, Canada, accounting for 9%
of total supply.
Other producers include Russia, Cuba,
Australia and the Philippines.
In total, cobalt is mined in 14 countries.
China Molybdenum has made recent
moves to secure long term supply for its
domestic battery industry by acquiring
Freeport McMoran’s Teneke mine in DRC
for $2.65bn and with it 20,000 tonnes of
additional cobalt capacity.
The move puts China to the number one
spot for countries that control the most
cobalt supply, leapfrogging both Canada
But it's still nebulous how 59% became 80% (or 85%).
As noted in the other answers, there's probably a confusion in some sources between "rare minerals" and (IUPAC) "rare earth" metals. Cobalt has been called the former (i.e. "rare mineral") in another Guardian piece, an article headlined by its staff this time.
So what about lithium batteries and rare earths? There's an article in The Bulletin of Atomic Scientists that probably clarifies much of the confusion(s). Basically lanthanum (a rare earth) was used is some old batteries:
Even in 2015, China accounted for more than 80 percent of global rare-earth production but also for about 70 percent of rare-earth demand—not an unreasonable balance. [...]
Some hybrid cars, like my 2001 Honda Insight, used nickel-metal-hydride batteries containing lanthanum [a rare earth], but those are now largely replaced by lighter lithium batteries, which typically use no lanthanum. (Both kinds of batteries are also recyclable, and infrastructure for recycling is emerging.) Tesla’s market-leading lithium batteries, like its motors, use no rare earths at all. Non-lithium batteries and potent potential substitutes for batteries (notably graphene ultracapacitors) are also emerging. [...]
Some writers claim that lithium, a light metal occasionally confused with rare earths by novices, is too scarce to support a big electric-car industry—a surprise to firms with vast lithium deposits in diverse countries.
This article does not specifically say that cobalt can be confused for a rare earth, but since lithium can... Furthermore, a blog on the site of the Union of Concerned Scientists says that the confusion can include cobalt:
Specifically, the use of lithium, cobalt, nickel, and other metals that are part of an EV lithium-ion battery pack has raised red flags about the poor human rights and worker protection records in the countries where these materials are mined.
A lot of these warnings have been incorrectly categorized under “EVs and rare earth metals.” Though neither lithium nor cobalt are rare earth metals, and rare earth metals aren’t nearly as rare as precious metals like gold, platinum, and palladium, there are important issues surrounding the production of lithium-ion batteries that must be acknowledged and addressed.
And if you need more proof that the confusion is likely, an article in The (Toronto) Star calls cobalt a "rare metal".