Can a planet (gas giant or otherwise) have rings not around the equator, but instead rings running from pole to pole? Think of Uranus' rings, except the planet isn't on it's side
They can have rings not aligned with equator, problem is how they came to existence.
Normally rings are formed together with planet, from the same particle cloud, partially condensing into the planet itself, but also partially remaining afloat in orbit and slowly trimming their orbits till they become the ring(s) (see this).
Given this formation they obviously share axis of rotation which is exactly the same as the original cloud.
OTOH if rings came from disintegration of a "captured" satellite (like Triton, orbiting around Neptune in quasi-polar orbit) then it could happen, but it's very unlikely.