It might be *[Spirit Animals][1]*. 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.

[![Cover of first book][2]][2]


Partial summary from the Wikia page for [the first book][3]:
> 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.

  [1]: http://spiritanimals.scholastic.com/about
  [2]: https://i.sstatic.net/CTgY3.png
  [3]: http://spiritanimals.wikia.com/wiki/Wild_Born