As suggested by another commenter, that's a false dichotomy. I'd argue there's lots of room for innovation and refinement here.
Manual or algorithmic identification of family members is one option, but I think there's something more to be gained here by understanding the behavior just a bit more abstractly... Shopping moods? Modes? Targets? Tasks? Something, possibly – but I'm only guessing here. The real power could come from actually working with/doing user research and prototyping.
I'd love the chance to work on such a project, actually – I had a chance to work with data from a study on the online shopping habits of mothers a few years ago and there's lots of interesting angles to possibly explore, IMHO.
Honestly, I have to be nervous logging into my personal account with other people standing around. The "Recently Viewed" and "Recommended for You" are full of things I'd really rather my family members and most of my friends see.