I'm having the same issue with Prestashop 1.6.1.6 and, so far, only on Google Chrome.
I searched the source code, and there's no mention of the non-unique ids anywhere, except for those button elements.
I removed them and got no errors so far, except Chrome shows no warning anymore.
Still, the behavior is very different between Firefox and Chrome.
On Chrome, I can only update a product the first time. After that, whether I use "Save and Stay" or just "Save", the changes are not saved and "Save" does not go back to the product list.