Untwisting stuck salt shaker

I would go with the first answer regardless of the problem of the salt bidding the container and I strongly suggest to avoid applying cold or ice-water as you suggest. Every container that I had problems opening was way easier to open after immersing it in warm or hot water and then drying it thoroughly. This is personal experience I am not sure of the physics behind.

Warm water will help dissolve the salts in the threads.

If it were a metal top on a glass container the difference in the thermal coefficient of expansion could be used to help loosen the lid. Because the lid and container are the same materials, You won't get much help from the thermal expansion or contraction.

Soak overnight in white vinegar. I tried everything to get the lid off my sterling silver salt shaker which was stuck on by corrosion - hot water, ammonia based products and even pliers which were in danger of causing permanent damage. It wouldn't budge. Online forums also casually suggested using white vinegar. And magically that did the trick! I soaked the top of the salt shaker (upside down in a very small jug) in white vinegar overnight and hey presto, the next morning is easily unscrewed.

Definitely white vinegar but ten to fifteen minutes is plenty. The vinegar can discolor the metal of the shaker or take the plating off. This worked perfectly for me. I had two that were utterly stuck. Like never coming off. And the vinegar did the trick. You can see the corrosion on the top one.