Before a child can apologise, and mean it, they must realise they’ve done something wrong - a concept pre-schoolers don't generally grasp because they have a very underdeveloped theory of mind.
"The presumption that others have a mind is termed a ‘theory of mind’ because each human can only intuit the existence of their own mind through introspection, and no one has direct access to the mind of another." - Wikipedia
Pre-schoolers struggle to understand how others are feeling (empathy), particularly if it differs from what they themselves are feeling. Empathy is one of the last social skills to develop in children.
It's reasonable to expect a decent amount of empathy once the child starts school, but toddlers and pre-schoolers are notoriously lacking in empathy skills.
Why does empathy matter when it comes to saying “sorry”?
Sorry implies that the child feels bad for what they have done, and in order to feel bad they have to understand how they have made another feel – which they don’t have the tools to do yet.
‘Sorry’ Doesn’t Teach Empathy
Asking a pre-schooler to say sorry doesn’t teach them empathy (in fact, it may just teach them that “sorry” is a magic word that gets them out of trouble).
We don’t teach a pre-schooler to write by handing them a pencil and paper and asking them to write an essay. They don’t have the available skills yet. Instead, we start slowly, letting children make marks with pencils, crayons and paint and we build from there.
If you are at a playground and your child pushes another and makes them cry, don’t ask your toddler to apologise. Instead, apologise to the other child yourself (and the child's parent) because you are genuinely sorry.
This is a great role model (demonstration) for your own child.
Then, speak to your own child in a quiet area where you explain, simply, that the other child was crying because it hurt when they were pushed. This is the way to help develop your child's empathy skills.
The chances are they will still shove the next time though, because that's what two and three year olds do, but if you keep repeating this process each time, you have a much greater chance of raising a truly empathic child.
One who sincerely means it when they say “sorry” when they are older.
When it comes to teaching children how to apologise and make amends, guidance and patience from you is much more valuable than “sorry”.