Alt text is a part of the HTML not the image. I completely see why you would want to associate it with the image and have it follow that image wherever you use it. But since alt text is intended to provide a description for an image in a specific context, this sort of universal alt text is beyond the intended scope.
Appropriate alt text changes from context to context, even for the same image. A picture of a nightingale in a 19-Century Romantic Literature course might include something like: "The nightingale, considered to be the bird of love, for its beautiful song" The same image in a a biology course might have the alt text: "The common nightingale, rufous nightingale or simply nightingale (Luscinia megarhynchos)"