Children can with lots of practice from age 1 can write their first and last name by age 3. Most do not because there is not an adult working with them on that skill. Children go through several stages of writing before they end up with a product that looks like their name. Many children are well able to write their first (and possibly last) name by age 4.
The usual progression of name writing looks like this. First, they are scribblers who make “chicken scratch” stokes. Nothing makes sense. Next, they start making some shapes or scratches that they may begin to name. Sometimes what they say it is doesn’t at all look like what they produced. After that, they begin to scribble with some directionality. It might be straight lines, squiggly lines or zigzag things. Additionally, they start making things that might look like a letter. These are symbolic or mock letters. Finally, they start writing their name, but they might repeat several letters, miss a letter, or write the letter with extra lines or something. Somehow you know, this is their name in their own signature style.
Sometimes adults write a child’s name using all capital letters or maybe insert a capital letter somewhere in the child’s name where it does not belong. Beware of this error. Children will model what they see. Make sure that you do not accidentally use uppercase letters where they do not belong.