There is both a light side why we love children and animals so much, as well as a shadow side. Let me illustrate the light side first.
When children come into this world they are completely pure and blank slates, devoid of all the socialization constraints and conditional programming that we as adult have. They have no fears, limiting beliefs, repressed emotions, because they have not been taught them yet. They are therefore purely themselves, completely aligned being, following their heart desires exactly, without restraint. Of course as they grow and get older in society they learn all sorts of limitations, beliefs, prejudices, dogmas, etcetera which are placed on them by their family, their education system and society in general. They thus lose the ability to be themselves purely over the years, and they become fitted versions of themselves that is more acceptable to the social setting they grew in, at the cost of not being aligned with their highest potential, with not being fully themselves and following what their heart desires. This is adulthood, for most of us. It is no wonder that, as we watch the inherent freedom shine within a child, their natural spontaneity, their genuine honesty and truthfulness, we can't but feel attracted to what we ourselves have lost, and inspired by their ability to be their most truthful, highest version. This inspiration is the light part that attracts us to children so much. For similar reasons we are attracted to animals. Animals, being devoid of social constructs, and any form of socialization and education, are also grow as free beings. They are exactly themselves. This can greatly attract us to them, especially if we have a pet animal.
But let's look at the shadow side now. Although children, when they come into this world, are completely free beings entirely in alignment with themselves, there is one major caveat. They are not really free in the sense that they depend on the environment they come into. They are in fact completely dependent with their lives and thus survival on their caregivers and educators. They themselves are not able to support themselves, feed themselves or take care of themselves in any way when they are born. This puts them in a majorly fragile position. This places enormous power over them in the hands of their caregivers.
Now, if we lived in a society were parenting was a carefully learned skill, and where adults were conscoinsly aware of themselves and their issues, and thus capable of not passing them onto their offspring, that would be all great. However we live far from such a society. Currently, in the state of our world, parenting requires absolutely no skill (there is literally nothing preventing anyone, from the most complete idiot to the most sadistic psychopath can be a parent), adults in our current society are for the most part far from being fully conscious and self-aware being that have dealt with their issues, and indeed for the most part these issues are passed onto their offspring.
This means that when people decide to have children, they often do so both from a light motivation as well as from a shadow motivation. The light motivation could include anything form wanting to experience being a parent, to wanting to share their love with another human being. The shadow motivation could include the need to fill some kind of void in their life, or the wish to have a second chance in life by giving their children what they didn't have, i.e. by using their children as a make-over to their life, or graver shadows such as the need for gratification of the ego. Unfortunately i would like to say that these motivations are rare, but the reality is that they compromise the great majority of motivations why people in this world have children.
Given the shadow motivations why people have kids, combined with the inherent fragility and utter dependence of children on their caregiver, a pretty gloomy scenario is formed. In the best scenario a parent will project their personality onto their child, including their desires, needs and wishes, not recognizing the child as a separate and independent being. They will for instance expect the child to become what they want to be, or to do what they like to do. They will impose their value system on the child - including what is wrong and what is right, what is acceptable and what is not acceptable - instead of letting the child chose their own. They will impose their belief system onto their child, and so on. In the worst case scenario a parent with more or less serious issues will abuse their child in various ways.
When we talk about child abuse often what comes to mind is a parent beating a child, or molesting them. Although these scenarios are both still extremely common today, there are many more forms that abuse can take and which are much harder to detect. Neglect fro instance, is an invisible type of abuse because it comprises all sorts of needs, desires and wants of the child being unmet - both psychological and physical. Psychological abuse is not necessarily limited to violent and abusive language, it can take the form of invalidating the child's emotions and feelings. All these are serious forms of abuse which unfortunately are very common. Most parents are completely incapable of sesing the abuse they do on their children, for they cannot see beyond their own issues.
Being completely dependent on their caregivers, abused or misused children in all the above cases, have little choice. Unless they are lucky enough to find external support that will help them develop healthily, for the most part they can only resort to psychological defense mechanisms, and this is where all our issues begin. Unable to change or escape from the abusive situation, children must create a system to live within it while keeping mentally sane. To do this they internalize all sorts of belief systems, oppress all sorts of feelings and emotions, and create an internal narrative that justifies the situation, normalizes the behaviour, and lets them believe that what is happening is loving and is ok, while still managing to grow up. This is how trauma is created.
With animals of course the situation is similar. Animals, being themselves completely defensless and dependent - even more so than children as they do not posses the ability to speak, nor do they have solid laws to defend them - must adapt to any sort of situation. We see very commonly the scenario of the dog that is beaten by its owner, and then jumps and celebrates him in return. A kind of animal Stockholm syndrom that is the only strategy of survival the creature has in the scenario. People do all sorts of horrible abuses on animals, and animals must adapt.
So, to summarize, the shadow side for liking children and animals so much is the fact that - unlike adults - they cannot defend themselves from us. They cannot rebel or hurt as back and tell us how what we are doing is wrong. They have no choice but to accept and to love us for who we are. And this for a lot of people feels like unconditional love.