The mouse evolutionary strategy is to breed extremely prolifically, so that many mice can be killed and the species will still survive. Such a strategy, as with most strategies, requires a balance, which humans have interrupted. We have tipped the scales in favor of mice by giving them nice warm places to survive through the winter. There are now way more mice, as there are more cows, chickens, sheep, pigs, rats, pigeons and other animals that humans create favorable breeding grounds for than there would be, absent humans.
To me, morality calls for eliminating as many excess mice as possible. I think it is immoral to save mice. It hurts other people by encouraging the spread of disease and filth. It is not even morality when people want to save mice. They are, in fact, giving up their humanity when they fall prey to their unreasoning swoon to cuteness.
That swoon was designed to help us love our babies; however it does spillover and when we see certain physiological shapes, we are often touched. But make no mistake, there is no reason for this so-called “morality.” It is a side-effect of the “morality” that keeps us from killing our babies. Which isn’t morality, either. It’s just built into us via evolution.
What we are talking about is also called “empathy,” as @thorninmud pointed out. But I don’t think we should get very high-minded about it. Like I said, it’s built into it, but it isn’t reasoned. We come along later on, and try to build reason around it—perhaps. Mostly we just throw words like “morality” at it, and make it sound like we are doing something worthy and hope no one will actually think about it.