Animals in Norse culture have a spirit and soul, the same as humans do. The concept that only humans have a soul is indeed entirely Abrahamic and is abhorrent in Norse culture.
As a result, when it comes to the afterlife, animals such as dogs and cats have a similar standing as humans, and more specifically their owner. The pets (or working dogs) will therefore eventually re-unite with their dad wherever he ends up when it’s his time to pass (for more information as to where you will likely go in the afterlife, see “Valhöll”).
But until then, pets would normally find themselves in the Lundar area (peaceful wilderness) where it is always summer, where there’s a permanent all-you-can-eat buffet (which pets will inherently benefit from), and where they can chase all sorts of things!
In a world of shit. This puts a smile on my face