I am starting to get into frying again. I was wondering if it is ok to store the oil after it has cooled down in its original container or if there is a better option for storage.
If the container is labelled with one of the standard logos that mark an unconditionally food safe container, there should be no concern: Used peanut oil is still food, so a container specified to deal with food should deal with it too.
If not, assume that the container has only been designed to be safe with the exact, unaltered ingredient shipped in it (oil and plastic chemistry can get very complicated, some edible oils WILL dissolve some plastics).
The best option will always be an airtight, food safe, glass, stored in the dark container well filled.