The only word I can like 'grænsen' to is 'granitsa' (граница) in Russian. Perhaps an Indo-European languages link?
In addition to Miacomet's comment, consider the history of Russia.
The word "Rus" comes from "Rueriks", who were a norse ruling family.
It would make sense that the word for border comes from the ruling family's language.
I think this sentence is not up-to-date anymore. The Danish border was open...