Many businesses just start and are in existence. Unless needed for a particular type of business such as the professions those in self employment only have to register with HMRC within three months.
The business owner normally chooses a name or just uses their own name and off he or she goes. They advertise, they do the work and are paid. They keep records of sales and purchases.
Self employment seems easy but there is a downside. You are personally liable for any debts you incur through your business. If you cannot pay your business debts your creditors can make you bankrupt with all the nasties that that involves.
Also you do not own your Business Name unless you make it unique in some way and register it as a trade mark. You can however take steps to prevent someone pretending to be you. If they do not do that there is not much you can do in general terms.
There is a lack of regulation in self employment apart from any regulations that relate to your specific industry. As a result you only have to deal with HMRC.
The bookkeeping is more straightforward in self employment . Initially with help you can do it all yourself saving you money.