It's not shown on the list of NATO members because it's not a separate country. Other islands like Bornholm and Zealand are obviously also not shown, because they are Denmark. So is Greenland. They are not independent, though both Denmark and Greenland has worked towards that in many years.

Greenland was part of NATO since the beginning, and you could even argue that it was before that, because the collaboration between Denmark and USA on military control of Greenland predates NATO.