The poem is about
geographical places with 'duplicate' names. Thanks @OmegaKrypton, @Evergalo and @AHKieran for some of the answers/hints.
One of us – largest outside polar regions;
Sahara desert - 'sahra' means 'desert' in an Arabian dialect. (found by @AHKieran)
another one's island's shared with Indonesians;
East Timor - 'timur' is Malayan for 'east'. (found by @Evargalo)
a third one is found at the edge of the first,
a break from the dryness to quench people's thirst.
Lake Chad - 'chad' means 'large expanse of water' in a local language. It lies on the edge of the Sahara (see first line) and it's freshwater so it's drinkable. (found by @Evargalo)
One of us separates Swedes from the Finns;
Gulf of Bothnia - 'botn' is Old Norse for 'gulf'.
one – lake with skiing, casinos and inns;
Lake Tahoe - 'tahoe' is Washo for 'the lake'. (pattern found by @OmegaKrypton)
one's a dry place near where Mongol herds graze;
Gobi desert - 'gobi' is Mongolian for 'waterless place'. (found by @OmegaKrypton)
one flows where born was a writer of plays.
Shakespeare was born in Stratford-upon-Avon - 'afon' is Welsh for 'river'. (pattern found by @OmegaKrypton)