BTW, I just checked and the Glypish example, does in fact, use ID's
Indeed, it does. It's still wrong.
It isn't documented as the proper way to do things, though. It will only work reliably with IDs if those IDs are used on one and only one element on one and only one page of the site.
As long as those same IDs are not used anywhere else on the JQM demo site (and I suspect that's the case) then technically it's correct. But it's poor practice.
In your case, since you do duplicate that navbar on multiple pages, it will not work in certain browsers. There is no guarantee as to how a given browser will handle the case of multiple elements with the same ID, because that is not valid HTML and the specification does not say how to handle this situation. So, there is no consistency between browsers.