Even though there are no moral facts, moral laws, or inherent moral duties or obligations, moralization, that is, belief in moral values or norms and the attempt to live in accordance with those norms, is inevitable among humans and perhaps among other highly intelligent animals. All people everywhere have notions of what is right and wrong, generally seek to live consistently with those notions (at least in principle), and generally expect others to do the same.
The above theses are difficult-to-prove and difficult-to-falsify empirical claims. They could only be proven or disproven by polling a very large number of people, and even the results of those polls would only be reliable if people were honest. However, I think common human experience across all cultures is sufficient to warrant assent to these theses, even if it is not based on rigorous scientific testing.
To test these theses on animals, it would be necessary to observe animal behavior and infer the cognitive processes occurring in the animals in question based on their observed behaviors. While such inferences would be speculative to a large degree, they would still likely reveal general patterns of behavior that could properly be called moral.
These theses, assuming they are true, are nonetheless not proof that any particular moral beliefs/norms are true or valid. If true, they merely describe how people do moralize, not how people must or ought to moralize.
It is no doubt true that in practice the vast majority of people behave in ways that deviate from their moral norms/beliefs some of the time. For this reason, it is never possible to assume or infer that the fact that people do behave a certain way means they believe they ought to behave that way, or at any rate would believe they ought to behave that way if thinking about the situation dispassionately.
In addition, consistency and its opposite, hypocrisy, are themselves not required or forbidden, respectively, by any observable properties or facts of nature, nor are they necessarily perceived to be required or forbidden, respectively, by sentiment, a priori reasoning, or intuition. However, in practice most people do aim for consistency and judge inconsistency negatively. In other words, the requirement of consistency is an integral part of the way most people moralize in practice, at least in the contexts with which I am most familiar, but there is no objective or universal duty or basis in nature for treating consistency as a binding requirement of morality or moralization.