This is wrong. Svalbard is Norwegian, but has a special status

Svalbard is, according to the Svalbard treaty a part of the Kingdom of Norway:
The High Contracting Parties undertake to recognise, subject to the stipulations of the present Treaty, the full and absolute sovereignty of Norway over the Archipelago of Spitsbergen

There are however some restrictions when it comes to taxation and establishment of military bases, and Norway has to let individuals of all signatory nations have equal rights to work, mining etc. Norway however can establish national parks and introduce legislation for Svalbard, as long as it affects all nationalities equally.

As all signatory powers have equal right to exploitation of natural resources, there has been a significant Soviet and later Russian presence on Spitsbergen (Pyramiden and Barentsburg), but this does NOT make Spitsbergen partly Russian.

The reason for any passport controls is that entry requirements are different for Svalbard and Norway.