I have a close friend who lives in rural Ireland. That's not in a town, but on a small farm in a general area of about 15-20 other small farms. His address is just the name of the general area and the county. The Irish postal system is clever enough to know that the person with this name, in that general area lives in that house. In fact, he probably still isn't even really classified by his name, rather as the husba…
The problem is if someone in Ireland tries to order something from a company in another country, where the computer system isn't set up to handle totally arbitrary addresses.
Even UK addresses look pretty anarchic to someone in America, where every address has a street name, a house number, a city, a state, and a zip code. UK addresses will have a postal code, and a city name, but after that there's no telling; it'll frequently list smaller and smaller towns and then some hamlet, and not have any kind of street or house number.
If you design your address-storage system to just handle two arbitrarily-long "street" lines, plus a city and postal code, that should handle everything in the UK. For Ireland, just don't require a postal code. The problem is when programmers try to place restrictions on the address to ensure it has "correct formatting". Don't do that; if you need to verify the address, there's APIs you can subscribe to in different countries to check address correctness. Doing it yourself is how these problems happen.