The alt text is what gets used for those people who can't see the image. Unless the image is being used for a different purpose in each page it ought to have the same alt text as that is what people who can't see the image will see or hear instead.
The only time that an image should have different alt text on different pages is where the purpose of the image is different. For example if you have a photo of you and your dog and one page is about you then the alt text might say 'me with my dog' while the same picture on the page about your dog might simply say 'my dog Fido'.
Whatever alt text you use should convey the same meaning as the image would.
I cannot conceive of a situation where the same image would identify which page you are on for more than one page. If the image is supposed to identify the page then a different image would be needed for each page.