What does this mean? I think it is a proverb. Something do to with god?
In my research, is it similar to 江山易改，本性难移?
Yes, similar to 江山易改，本性难移. It refers to someone who cannot change their bad habits, like dog cannot change their behaviour to eat shit.
For example, you can say a thief commit a thievery again and again, then you can say: 這小偷老是偷東西，真是狗改不了吃屎。
It is used more often in a colloquial language, and less severe than 江山易改，本性难移. When someone uses 江山易改，本性难移 to another person, it is often with the emotion that they have been let down massively, and may even be angry/disappointed with the person. Whereas 狗改不了吃屎 often may just be complaining, but don't really care if they change or not.
I'm Chinese. The differences between the two phrases are: 1. 江山易改，本性难移 is used more neutrally while 狗改不了吃屎 is definitely more negative for complaining. 2. 江山易改，本性难移 is often used in writting, whereas 狗改不了吃屎 is informal and often used in spoken lanaguage.