Yes, both can go rancid, but the animal product is less inclined to do so at room temperature. Quite apart from rancidity, there is the problem with low essential nutrients in seed oils. This is a rather convincing presentation just a few weeks ago:

(I have no particular business, investment, managerial, or personal interest in the organization, nor with the presenter. I just watch these videos in an attempt to get a good handle on the latest thinking and findings.)

Here is a cardiologist, Dr. Nadir Ali, from Texas, who gives a short argument for higher cholesterol levels, especially when one watches the carbohydrates: