With the new cats in 1.14, I think that you should be able to pick up your cat by shift-right clicking on them with an empty hand. They would appear as an item (called "Cat", or the name given to the cat with a nametag). If you died, they would appear sitting down next to you where you respawn, and not drop like other items. Because who wants to walk 100 blocks to get their cat? That's a devastating situation.
This would also be useful for moving your cat without a lead, because from what I know, cats don't follow you around.
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