In Germany you'd need to license as sole trader ("Gewerbeschein") costing about 25 Euro, depends on city/county. You're still an individual and keep your personal tax number, but now indicated you plan to make money besides being employed. Same procedure if you want to do freelancing. Or write newspaper articles for money. Or anything that generated income. So if you're a freelancer it would be already covered. Any accountant can walk you through the steps. Invoices will carry your name and home address. You're fully liable with your money.
Starting a limited business is separate. Then you need a separate bank account, register with the tax office, get a separate tax id. Depending on type of business you might need to bring in money (25.000 Euro for a German GmbH) or you need two founders. Year end taxes are more complex. Disolving a company is a pain.
VAT is also separate. Freelancers in Germany for example are exempt until 17.000 Euro, the limit is even higher in the UK.
So legally you probably need to register. Since you're already fully liable even when registered some people delay it, even months and then backdate. Not ideal but afaik there's no fine. I delayed by 5-6 months, tax office didn't care.
Individual customers won't notice, when is the last time you looked at the imprint of a website? (I do, but I might be the exception). B2B customers probably care. And larger account will already wonder "will this website still be online in 6 months" regardless.