I've been looking at photos of Caroline's necklace and at first glance, it does look a bit different than the tiaras worn by Tatiana and Princess Charlotte. But I'm not sure if it's because it's a different piece, or because it's being worn as a necklace. I imagine it hangs/displays differently when not in it's frame.
I wish we knew more about the tiara too. I think the only clue we might get is when Charlotte or Pierre get married. If either Charlotte or Pierre's bride turn up in a fringe, then we could assume that Tatiana's tiara is part of the Grimaldi collection.
My family have none either. But that's probably good because if they did, I might end up wearing it every day.