By studying the words and what their place is in human society, people believe that this phrase refers to the insignificance or artificiality of human status distinctions. In other words, a person may call themselves a King or Queen but that does not make them any better in the eyes of society. This interpretation also means that an inferior is not restricted in what they are allowed to do in front of their superior. Every person has a right to approach another, even though they may be their superior or their social statuses may be different. This phrase can also serve as a reminder that we are all equal to one another regardless of social status, education, age, gender, religion or ethnicity.
Another possible interpretation of this phrase is that looking at a Queen is quite a rebellious thing to do, so by the cat looking at the Queen, the phrase is suggesting that rebelliousness and disobedience are not just human traits as they can also be seen in animals, in this case, felines. If the expression dates back to early Britain when cats were simply kept in castles for catching mice, then it suggests that they were the lowliest servants whereas the Queen was the highest authority. It can also be a reminder that authority, in this case the Queen, is there to be questioned by its subjects, the cat.
Or, completely contrary to the above meaning, the phrase could be suggesting that cats, who were once hailed as Gods by the Egyptians who believed they were the Guardians of the Underworld, have privileges that we humans do not.