In Ultimate Magic, this bloodline is introduced as a variant of the Draconic bloodline. So far so good.
Where I am unclear (and I could just be me) is are we dealing with actual linnorm (Ice, Tor, Crag, etc.) or a more Primeval Dragon variant?
In other word, do I choose Bloodline: Linnorm(Ice) or Bloodline: Linnorm(White or Silver)
You choose sorcerer (wildblooded archetype) with the Draconic (chromatic or metal dragon type)/Linnorm bloodline. There is no separate Linnorm bloodline for the specific types of linnorms.
Yeah, you still have the normal Draconic bloodline, full stop.
The archetype changes some powers resulting from that bloodline,
but you still have and count as having the normal Draconic bloodline "as such",
just as an archetyped fighter still is a fighter even though they swapped some normal fighter abilities.
So to your question, the Wildblooded Linnorn (sub)archetype isn't really specific re: which type of Linnorn.
Choosing the standard Dragon type whose retained BL powers line up best with your preferred Linnorn type makes sense,
the archetype basically leaves that up to you to implement, i.e. there isn't specific fluff to match mechanics there.
The flavor is easy to tweak to suit your needs, because the only mechanical impact of any dragon is the energy type and shape of the breath weapon.
cairn linnorm = cone of acid = green dragon
crag linnorm = line of fire = brass dragon
fjord linnorm = cold = silver or white dragon
gare linnorm = cone of fire = gold or red dragon
ice linnorm = cone of cold = silver or white dragon
taiga linnorm = electricity = blue or bronze dragon
tarn linnorm = acid = black, copper, or green dragon
tor linnorm = cone of fire = gold or red dragon