In the Norwegian constitution from 1814 it states that Norway is a christian state based upon the christian faith. However, in another paragraph it directly states that jews are not allowed into the kingdom of Norway.
If I'm not mistaken, this was a common practice at that time, but isn't it a bit contradictory, considering that jews in Christianity are God's chosen people?
The issue is that in Christianity, "chosen people" doesn't mean what you think it does precisely. Quite a lot of Christian theology has historically held that Jews were the chosen people, but lost that by rejecting Jesus. This view has most famously been articulated by St John Chrysostom, but there are similar concepts in other texts (including the Christian bible I believe, but I'm no expert in that).
As you noted, this was far from the norm in Europe during that era (though this is at the tail end of that). Medieval Christians didn't see this as a contradiction. Jews were the chosen people, but lost this chosen-ness by renouncing Jesus. If anything, their direct rejection of Christianity by being previously chosen made them more reviled, not less.