My daughter was born with my (mother) last name, but when I went to family court to obtain child support from my daughter's father (we were never married), the judge stated that since she was under the age of 5, her last name would have to be changed to her father's. He never legally changed the name and I did not as the castodial parent. Since it was never changed legally, what do I need to do to make sure she does not have a issue with her last name in the future. Do I need to attest the court order change formally? If so, how do I do that?

Yes, never heard of a judge requiring it, even if the father had wanted it, if you objected most judges would not even order it. So something really funny sounding with what this judge said.

Next is this actually in the child support order ?
If it is in the order, then the father could go and demand it done at anytime, or perhaps father assumed you did it?

But yes, if you want to not change the name, or have it a issue latter, you need to get the issue back before the judge