Ive encountered this a few times.
Sometimes it's to stay under services radar, so for example people who arent here legally arent going to make themselves known to the authorities. People who are avoiding social services, its not uncommon people who have children taken away previously and who believe this child will be removed to try to hide the pregnancy not seek medical care, and try to avoid any form of registration. It also means they can try and hide from the usual safeguards eg people noticing that a child born is never registered at school. Sometimes its desperate women trying to hold on to their baby.
The second group which is more common than youd expect is the freemans. They believe that when you sign a birth certificate you make the child property of the state, and that you are signing a contract to abide by laws.
This is an example at court where the father explains his beliefs
www.theguardian.com/uk-news/2019/jun/23/man-refused-to-register-sons-birth-high-court Searches about freeman of the land uk, and sovereign citizens will bring up the same things from all over. Youtube is very much a wormhole of people being arrested and shouting that they havent entered into a contract with the police, and arent beholden to laws.
www.thesun.co.uk/news/8024005/conspiracy-theorists-parking-tickets-debt-uk-laws/
fmotl.com/
They attempt as adults to have themselves deregistered sometimes, but by not registering the child they believe they are exempt from most laws and taxes. Its an odd rabbithole based on maritime laws, and concepts around common law.
Its suprisingly common, and gets thrown up all the time. My local council for example has a page on it to say you must still pay council tax for example.
I used to work in a court and saw many people attempt to use it to say they had no jurisdiction. It never worked in my knowledge and i dont believe there has ever been a case that was won solely on that arguement (i did see cases, where the case fell apart due to other reasons eg. Errors in police investigation, where the evidence was lost or for example the parking ticket photos werent clear enough where people attributed that publicly to the freeman stuff).
It has a deeply explotive side where people asking for help with debts, legal troubles, fines, parking tickets etc on facebook are encouraged to renounce their citizenship, and sold a complete load of lies that mean they will be expects. Facebook groups with thousands of followers, youtube videos etc will all tell them to disengage with the system, often for a price (Theres lots of people that for a few hundred pounds will give you a certificate and tell you that your no longer a citizen and now exempt) then offer zero support when it goes tits up.
In the us for example theres cases where people have paid hundreds of dollars for people to share their knowledge, and give them certificates but then not been advised this impacts their ability to claim welfare.
As described above, there are ways to get round not having a birth certificate at birth, that usually involves getting one anyway designed to support kids born into situations they have no control of. However as someone whos parents didnt register her properly by accident, its a bloody massive pain in the arse to sort, and having basic id (often the only id you have as a child) is essential for lots of things eg passports, setting up bank accounts, applying for jobs etc. Once you have other forms of id, then you rarely need a birth certificate but often you need a birth certificate to get those ones.