First, your children are not an extension of you. They are a reflection of you...sometimes. Ever heard something come bubbling out of your toddler's mouth that made you wince, realizing what it must sound like when you say it? It is part of God's design that parents would be the primary influence on their children. It is only when that relationship becomes a source of such pain that the child rebels against this hard-wired bond. Even then, they cannot get your voice completely out of their heads. It should be a sobering thought. It is for me.
Secondly, it is a testament to our ego-centric society that we would consider another human being to be reduced to an extension of us. Each child is born unique, with a combination of physical and personality traits unlike any other, and God has a plan and purpose for that life. They are nothing less than a whole and complete person, even while they are dependent on our protection and care.
Why is it such a big deal to consider your child an extension of yourself? The biggest danger inherit in this kind of thinking is that if you do not view your child as separate, you will likely not grant him or her the respect, privacy, and even common courtesy you would extend to anyone else. You will be more likely to barge into a room without knocking, take something without asking, or interrupt without apologizing. Most of us would never admit that we consider children to be second class citizens, but sometimes our lack of basic respect shows that we have allowed ourselves to treat our kids in ways we would never dream of treating someone else.
Remember, we have been entrusted with positions of great influence and power as parents. You will have a significant impact on the formation of your child's character, one way or another. But let's not confuse connected with same.
There is a sovereign space between where you end and your child begins.