I don‘t believe that these two effects are mutually exclusive or even independent of each other. Remember, these results are based on asking individuals about their beliefs and perceptions. These concepts are often developed independently of each other, by different scientists from different countries in different decades. Many theories in psychology appear extremely similar, and, at the same time, painfully difficult to reconcile.
Illusory superiority is often about abilities: Most people agree to a statement such as „I am a better driver than the average person" or "I am better looking than the average person".
To my knowledge, false consensus research is mostly about beliefs and social norms (such as „littering is not ok“ or „climate change is a hoax“). People may generally believe that others (i.e. the general population) has more similar beliefs than they actually do.
It gets interesting if you combine behavior that is considered „good“ and beliefs about that behavior. People might, for example, believe that they are more honest than others, but at the same time overestimate the similarity of their own and other people’s opinions about honesty. Strange, right?
Then there is pluralistic ignorance. It may happen that people actually incorrectly gauge others‘ opinions or overestimate others‘ abilities. Typical examples are classrooms where everyone thinks others understand everything, even though nobody does. As a result, everyone is nodding along and nobody dares to ask a question.
Or, and this may also fall under pluralistic ignorance, people may incorrectly estimate other‘s acceptance of something, such as pollution, or littering. Some articles suggest that most people are actually really worried about climate change and the environment, but wrongly assume others are not, which leads to few people being outspoken about their opinion. In a way, people believe they have a superior attitude and everyone else is inconsiderate, which brings us - full circle - back to illusory superiority.