I have two DC, one 4, one 7.
The youngest has always shown tremendous empathy from an early age. She is the child who will comfort you if you are upset, will ask you if you are feeling tired, will reassure you. It is quite astounding. When she plays role play with her sister she will actually cry if she is acting a role. I ask her if she is ok and she says. "Yes I'm only pretending" but she is really crying and I find it quite upsetting. It does happen more when she is tired.
Her sister leads the role play and is very controlling so she does have to concentrate for long periods of time.
My eldest child, in contrast is very sensitive and an anxious child. She picks up on other people's stress and does become quite difficult when stressed herself but I can't recall her showing empathy to others.
If her sister is upset there is no attempt to comfort. Other people's upset tends to make her angry so I believe she gets stressed by it but she isn't sympathetic towards it. She does not or rarely shows remorse if she hurts her sister or someone else. She does express empathy towards animals in the sense of not wanting to hurt or eat them but towards people she seems indifferent.
I'm sure these are both normal but different ends of the same spectrum but was curious as to other people's experiences of empathy in their children.
My eldest was never a huggy child either- although very clingy and only recently does she say I love you. Again my youngest is the opposite.