Let me start by pointing out that most of what is written in the website you linked has no basis whatsoever in the sources that make up the old Norse corpus.
Regarding owls (old Norse ugla), their mentions in the old Norse corpus are extremely scant. Rather uninterestingly, they appear in lists of birds and ship names in Skáldskaparmál.
However, a quite notable mention appears in chapter 21 of the Völsunga Saga, in which Brynhildr recites a poem about runes to the hero Sigurdr. She says (translation by Kaaren Grimstad):
Master mind runes
if you mean to excel
in wisdom among men.
Hropt writes them
and reads them;
he fashioned them first.
They were cut on a shield
for the shining god,
[...]
On bear’s front paw,
on Bragi’s tongue,
and on wolf’s claws,
Cut on eagle’s beak
and bloody wings,
on bridgeheads, too,
on merciful palms
and on pity’s soft path.
On glass and gold
and good silver,
in ale and wine,
on witch’s seat,
on men’s flesh,
on Gaupnir’s fierce point,
on troll-woman’s teats,
on norn’s fingernail
and the nose of an owl.
The same poem appears, this time recited by Sigrdrífa, in Sigrdrífumál, a section of the Poetic Edda. Notice how the other animals mentioned in the poem (wolf, bear, eagle) feature prominently in Norse myth and sagas. It could then be that owls held a more significant role in old Norse culture than what the surviving sources suggest, but this is just speculation.