So in US, you can change your last name as you could with your first name. So imagine as a father, you changed your last name from the one your family gives to you. Can you choose your child to use your family's last name not your changed last name?
No law in the US requires that parent and child have the same last name. It is usual that a child's name match that of at least one parent, but not required.
A parent can change his or her name, without changing the names of any existing children. Also, when a child is adopted, the child's name need not be changed to match the name of the parents, or either of them.
I have read of cases where a widow remarries, and takes the name of her new husband, but an adolescent child retains his or her birth surname. I suppose this would also be possible legally if it is the husband who changes name on remarriage, but i have not read of such a case.
I think, but I am not sure, that a child's name could be changed to a different name than the name of either parent. It may be that this would only be done if the child is old enough to understand and agree to the change.
Your legal name is your legal name and your child’s legal name is their legal name. For legal purposes you have to use your legal name.
When you name your child there is a convention that they take the father’s or mother’s (or both) last name but you can give them any name you like (subject to names the state restricts).
In general usage, you can call yourself and your child can call themselves anything they like - most people have different ‘handles’ in different circumstances.