Why not stylize it as typographic but store it as a "regular" double quote though?
One thing that makes computers more powerful than analog tools is that they are much much much more structured.
So while we should not take computer limits for granted I think that the opposite extreme is just as if not even more absurd.
Or, you know, you could just have separate characters for opening/closing quotation marks. ;-)
Yes, that's how ligatures work. Pairing across characters is the one I'm not sure I've heard of before in fonts.
Edit: Besides, how do you think these are typed on keyboards without separate keys? The software already detects and replaces them based on context.