I think your sister was stupendously rude - and also wrong. In the instance of the earrings then either is correct. But in fact I think that "those" is most appropriate when there is a choice on offer e.g. several pairs of earrings and you might point to one and say I like those. If there is only one set, then them is fine in my book.
Where are my keys? I am always losing them. I have a new pair of shoes - I really like them. My neighbours are a pain in the arse - I really hate them.
I was out with my sister and I have to say it wasn't the most successful evening I've ever had but that's by the by. It started when I admired her earrings. "I like them" I said. "Those" she said!! She corrected my grammar! The nerve! The cheek! So here's my question. Is "I like them" wrong? Really, really wrong? Or was she being a pain?