Q:

Where Mongol herds graze

One of us – largest outside polar regions;
  another one's island's shared with Indonesians;
  a third one is found at the edge of the first,
  a break from the dryness to quench people's thirst.  
One of us separates Swedes from the Finns;
  one – lake with skiing, casinos and inns;
  one's a dry place near where Mongol herds graze;
  one flows where born was a writer of plays.

What do we have in common?

A:

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)

A:

@Glorfindel has found the answer and explicited the second stanza, here is an explanation for the first one:
One of us – largest outside polar regions;

 @AHKieran pointed out in a comment that this must be Sahara Desert, where "Sahara" means desert in Arabic.

Another one's island's shared with Indonesians;

 Timor Island is shared between Indonesia and East Timor. But Timor/Timur also means 'East' in Indonesian. So this matches the tautological name pattern noticed by Glorfindel.  https://en.wikipedia.org/wiki/East_Timor

a third one is found at the edge of the first,

 Then this will be Lake Chad. From wiki, The name Chad is a local word meaning "large expanse of water", in other words, a "lake".

a break from the dryness to quench people's thirst. 

 Al-Ahsa Oasis in Saudi Arabia is the biggest oasis in the world. In Classical Arabic, 'Ahsa' means the sound of water underground.

A:

Partial Answer
One of us separates Swedes from the Finns;

 Tornio/ Torne River

one – lake with skiing, casinos and inns;

 Lake Tahoe (thanks @Glorfindel), a lake with the name that means "the lake", literally

one's a dry place near where Mongol herds graze;

 Gobi Desert

one flows where born was a writer of plays.

 Stratford-upon-Avon, birthplace of Shakespeare (thanks @Glorfindel again), where Avon literally means "river"

