From a former BITTERS specialist of 30+ years: "If it isn't EMBOSSED bitters or have an authentic label, it AIN'T a bitters". Now, that is just my opinion along with 95% of BITTERS hardcore. To my knowledge there has never been an authentic BITTERS found in that shape. That does not mean it isn't a Bitters.
My input is just enjoy & appreciate them for what they are no matter the contents. VERY nice appearing bottles.
I would consider them to be ketchup bottles but they could have held something else. Neither of them are common and an amber ketchup bottle is a very hard thing to find.