Hmmm, this seems like a bit of smoke and mirrors and diversion; doesn't the volume of an object/shape = the amount of liquid the object displaces (has nothing to do with inside world versus outside)
Topology, I was a math major, and I barely got it. I gladly took a “C”. Your real world example doesn’t apply Canuck. There are worlds of many more dimensions than we are aware of, topology is the study of shapes and objects in multiple dimensions. We operate in 3 and it becomes rather difficult to visualize when working with more.
So a beer bottle does not meet the definition of volume untill it is capped and the goofy shaped bottle is pretty useless in my mind. you can't fill a flask with liquid without venting displace gas within. Nothing new there. Be possible to fill it with sand. The definition of volume is new to me.
Seems I had a bong like that once. For a volume calculation your going to construct an imaginary plane over the opening, just like any right circular cone. Seal the open end, submerse in water and note displacement, then subtract volume of glass used to construct the shape. More than one way to do it.
Have fun with it. I know there are better brains than mine here. Just reminds me of my high school sciences teacher who always said "Physics is Weird stuff!"
Things in the physical world don't always operate in the manner we think they do. FEA (Finite Element Analysis) in structural engineering often seems to defy what we laymen mistakenly believe is happening in all sorts of objects.