I doubt there’s a law that requires having a name per se. But any number of laws and regulations require a name to do various things, like get a paycheck, own property, put a child in school, etc. Then there’s the practical side of things - unless you choose to live as a beggar or somehow scrounge a living in the wild, lack of a name will quickly prove a huge obstacle. Who’s going to cooperate in any facet of daily life with someone lacking a name?
The answer to any question of the form “Is there some kind of law…” or “Is it (il)legal to…” is invariably, “It depends on the jurisdiction.”
I can’t speak for Pierce County, Washington, but I do know that many countries in Europe have very strict laws about names. In Germany (and many other countries), for example, parents must choose a name for their child from an approved list. Exceptions are granted only in unusual cases, such as immigrants who wish to give their child a name from their own language.
Sweden famously had the case of Elisabeth Hallin and Lasse Diding, who refused to name their child in protest of one of these laws. They were fined SEK 5000 (USD 682), upon which the parents relented and named their boy Brfxxccxxmnpcccclllmmnprxvclmnckssqlbb11116. Unsurprisingly, the court rejected this.
Reminds me of the story I read somewhere about the man who got a vanity license plate for his car that said “NO PLATE”. He thought it was pretty humorous until he started getting deluged with parking and traffic summonses for all the tickets the police had written with “NO PLATE” as the plate number.
True and I am not finding an official answer on that. But I would assume the intent of the question is “can I be a functioning member of society and have no name?”
Since birth certificates are the underlying identification method for many, many points of contact with functional society I’d say the answer is no.
There are options for most of these things for people who don’t have a birth certificate but they are tedious and still require production of evidence of your birth at a certain place and location and not having a name on any of those things is going to make it pretty difficult. And then you’re still going to have to provide a name anyway (I’m pretty sure you can’t get a social security number not tied to a name, or a passport, or a drivers license).
So there may not be a law that absolutely compels having a name but I don’t think that means it is possible to not have one unless you live entirely off the grid.
In Japan there is a legal requirement to file the paperwork when a baby is born, in which the child is automatically assigned the father’s family name, if the parents are married or the mother’s family name otherwise. The baby is recorded in the family register (which is a Japanese system).
Japanese family courts have rejected some of the strange names which some parents have applied for, which, IIRC, has lead to a standoff for a couple of years, in which the government refused to pay for various expenses normally covered by the national health insurance.