That's not true for most uk postcodes - many in rural areas represent an area larger than even one town, so a full address is also required. They vary from representing one address to hundreds, it's not really consistent.
It does seem strange that postcodes are not unique to addresses but they were originally just zones intended to help routing. Eventually we will probably end up with one global address space with unique identifiers for addresses like ips and dns; it would clear up a lot of ambiguity while leaving the rest of the address to function as a human readable equivalent. lat,lon doesn't really work as addresses can be on top of one another.
Almost every rule you can think of will have many exceptions in the uk as postcodes are an ad hoc scheme covering areas of varying size - my block of flats has no number, many houses have a name, not a number, many rural houses don't even have a street name or a number, just a name and hamlet name, and many rural houses share a postcode but are not on the same street or even sometimes in the same village. So postcodes are not unique to a street, nowhere near it. Here a quick search for a rural location in Scotland sharing a postcode:
There happen ( by chance ) to be two addresses sharing a number and postcode there in the first 50 addresses, but that's the least of the problems for a scheme using no + postcode as a unique id I'm afraid!
There don't seem to be any properties in that list that violate the rule? The rule is house name or number and postcode. In the cases above, there are no houses that conflict on names or numbers that I can see?
Visit  and try DE4 4HA - addresses include "1 Council Square Brassington, MATLOCK, DE4 4HA", "1 Pleasant Cottage Miners Hill, Brassington, MATLOCK, DE4 4HA" and "1 Windyridge Red Lion Hill, Brassington, MATLOCK, DE4 4HA"
Try HD7 4PD to get "1 Moles Head, Golcar, HUDDERSFIELD, HD7 4PD" and "1 Prospect Place, Golcar, HUDDERSFIELD, HD7 4PD"
And of course there's the other direction: EC1N 8QX covers a bunch of flats in the same building, so one postcode includes "Flat G.7, Ziggurat Building, 60-66 Saffron Hill, LONDON, EC1N 8QX" and "Flat 1.1, Ziggurat Building, 60-66 Saffron Hill, LONDON, EC1N 8QX" - in other words, multiple properties have the same number within the street, and the same latitude and longitude. Also the building "number" has a hyphen, the flat "number" has a dot and can have a letter, and the building has a name too.
This is not a full list of addresses, but there are two no 10s there, and obviously as the postcode covers a large area with many streets, there will be some duplicate numbers, and many houses don't even have a number or street, and if you start to include house names as numbers, you will have duplicate names there too. Many addresses are as simple as rose cottage, village, POST CODE, and there might be many 'rose cottages' in that postcode.
It would be nice if it was a unique identifier, but the uk postcode is not, even combined with part of an address like 10, or even 10 high street, sometimes the village is also required to narrow it down. This might have worked for you on a limited set of data, but the assumptions are not valid across the uk.
>"The Local Authority will liaise with the Royal Mail to ensure there is no conflict with names of other properties in the same street or immediate area, before formally registering the name. If there is a problem, an alternative name must be submitted. In some cases, the Local Authority may explore the possibility of a house number being registered at this point, in addition to (or instead of) the new name. Once the change has been approved, the Local Authority will normally advise relevant bodies such as the emergency services. The same procedure applies for brand new properties which, for whatever reason, cannot be numbered (however, virtually all new properties today are numbered)." //
It may seem unlikely, but that's the way it works (see better examples in the michaelt post above). Often postcodes cover more than one village, and there are thus duplicate street numbers or names. Some attempt is made to avoid clashes for new addresses, but there are plenty of existing ones. You need more than a postcode and number to identify an address, sometimes a street and/or village is also required.
Not quite, in rural areas, most houses have a name. If a name is used instead of a number then the name must be unique. Obviously this doesn't affect those houses on normal suburban streets which have a number but also a name, in those cases the number is the identifier for the house.
The UK is pretty much an exception to the rule here though. In most other places in the world, including the US, post codes are not nearly as precise (source: worked on a real estate search engine that deals with addresses in 8 countries, including UK and India).
OP suggested a simple solution to an inherently complex problem in a complex domain. Doing what you suggested is like building a quorum protocol based on some intuitive idea about how a quorum should work and then fixing "edge cases".