Do my cats really know I exist?
Although there is a robust science of animal behavior, it's very difficult to determine what kinds of experiences or thoughts animals are having. In short, we can only observe their behavior. Philosophers argue that a "zombie" or a robot could also demonstrate exactly the same behaviors but have no mind behind them, which means that we can't infer the thoughts just from the behaviors. We can't measure the thoughts themselves. Even neural activity is just a proxy; they are not the thoughts. Frans de Waal's recent bestseller goes into great depth on animal emotions and suggests that there are no uniquely human emotions; I'm guessing he would think that your cat understands you as an agent and has loving feelings towards you.  https://www.nytimes.com/2019/02/25/books/review/frans-de-waal-mamas-last-hug.html