Mothers carry their children mostly on their left side, and apparently, wild mammals also --at least while fleeing their predators-- mostly keep their young on the same side as humans.

Moreover, many mammal Cubs prefer to approach their mothers from one side, and the reasons for this condition may be due to the fact that both halves of the brain regulate their control in opposite directions.
In mammals, the right lobe of the brain is responsible for processing social bonds and establishing relationships. This lobe also receives signals from the left eye. Some researchers suggest this may explain why human mothers and ape mothers tend to carry their babies on their left side. Because the facial expressions of the baby carried on the left side of the lap can be better observed with the left eye.
The study, published in Nature Ecology & Evolution1, investigated whether animal offspring also preferred to observe their mothers on one side. As part of the research, the team observed 11 different species of wild mammals from around the world, including horses, deer, antelopes, oxen, sheep, walrus, three species of whales and two species of kangaroos.
When the cub approached the mother from the back, the researchers reported whether the cub was positioned to the left or right of the mother. In 175 mother-and-offspring meetings, almost 11,000 locations were registered. As a result of the observations, it was observed that the offspring of all species tend to be positioned so that their mothers are on their left side. This way of positioning occurred at a large rate of three-quarters.
The observations also showed overlap with a recent human study published in Brain and cognition, which found that when children approach adults, they try to do so in a way that keeps adults on the left.
Being On The Left Is Vital
It was found that mammalian offspring positioned so that their mothers were on their left side had a better ability to follow the mother, thus increasing their chances of survival. For example, when baby whales and horses go for a ride with their mothers, the Cubs tend to move by coming into contact with the mother's body, reducing their chance of being accidentally left behind. However, if a threat arises, the roles are reversed.
According to the study, in normal situations, such as walking or breastfeeding, Cubs are positioned to hold their mothers on their left side, while in situations of threat, such as escaping from a predator, mothers position their cubs to be on their left side so that they can better observe them.
Human mothers, on the other hand, keep their babies on their left side, but this can change as the child grows and becomes more independent. In all studies conducted on mammals, the constant use of the right lobe of the brain in mother-offspring interaction indicates that this is an evolutionary advantage.
According to Lesley Rogers of the University of New England, if you have different tasks to perform, you can do these tasks more effectively if you separate both lobes of your brain for different types of processes. So it makes perfect sense to divide the right part of the brain into social behaviors.