No they are more like Pakistan, India or Australia and the UK's Commonwealth, since they aren't an overseas territory like the UK's Falkland Islands, but still share a common history. It's just a fancy title, the Danish Realm, but in reality the Faeroe Islands are an independent nation with links to Denmark, via a treaty with Norway.
They use their own, unique language, that shares some commonality with Icelandic, but only in written-form and the old-world language of Norn - of the Northern Isles of Shetland and Orkney.
Denmark, Norway and Swedish share no commonality with Faroese. Here's a breakdown of the regional history of the regional languages - Faroese takes its roots from Old West Norse, since it was derived from Icelandic, a Norwegian territory founded by Norwegian settlers.
The approximate extent of Old Norse and related languages in the early 10th century:
Old West Norse dialect - Red
Old East Norse dialect - Orange
Old Gutnish - Pink
Old English - Yellow
Crimean Gothic - Blue
Other
Germanic languages with which Old Norse still retained some mutual intelligibility - Green