It's a series of fantasy books in which the inhabitants of the world, from the time when they're children, get to somehow have an accompanying small animal that represents their soul or their personality.
A dæmon is a type of fictional being in the Philip Pullman fantasy trilogy His Dark Materials. Dæmons are the external physical manifestation of a person's 'inner-self' that takes the form of an animal. Dæmons have human intelligence, are capable of human speech — regardless of the form they take — and usually behave as though they are independent of their humans. Pre-pubescent children's dæmons can change form voluntarily, almost instantaneously, to become any creature, real or imaginary. During their adolescence a person's dæmon undergoes "settling", an event in which that person's dæmon permanently and involuntarily assumes the form of the animal which the person most resembles in character. Dæmons and their humans are almost always of different genders.
It might be Spirit Animals. It fits what you are looking for. In it, children at the age of 11 get to drink a magical mixture which summons a spirit animal. I loved it very much.
Partial summary from the Wikia page for the first book:
Enter the world of Erdas, where every child who comes of age must discover if they have a spirit animal, a rare bond between human and beast that bestows great powers to both. A dark force and forgotten lands, and has begun an onslaught that will ravage the world. Now the fate of Erdas has fallen on the shoulders of four young strangers . . . and on you.
It has children who have animals associated with their inner soul or personality, although it does not happen to everyone, and most of these are not small animals.