"Realize" is too strong a word. You're the only one who can verify that you're the soul who's staring out at the world through your eyes. For all you know, everyone else could be just biological automatons, golems.

Any leap beyond that is based on empathy. You have a soul, and you are human, therefore other humans could have souls too. It's a spiritual belief. Answers to questions that cannot be answered.

That's the standard Piagetian understanding of child development, yes. Humans do not start out with theory of mind, and are thus inherently solipsistic, but in most cases an understanding that there are other conscious beings with their own thoughts, goals and feelings develops between the ages of 2 and 7.

Developing theory of mind is one of the key milestones in child development.