One common theme is the loss of a cat. This could be through death, separation from their owner, or getting lost. Such themes often tug at the heartstrings as they deal with the pain of losing something precious.
Another theme could be the struggle of a cat in a difficult situation, like being sick or facing abuse. Reading about their fight for survival or their pain can be very emotional. For example, if a fanfic details a cat's battle with an illness and how it tries to stay strong despite the odds, it can make you cry.