A full postcode is often much less than a single street.
Picking something at random stick “SW15 6DZ” into Google maps and you’ll see it only covers 6 buildings (most are individual houses but some are split into flats). According to the Royal Mail address finder site there are only 12 unique delivery addresses that share that postcode. The Western half of that road has 12 or so full postcodes for only 100 houses.
A full postcode and one other bit of information can often be enough to uniquely identify someone.
If a US 5 digit zipcode is roughly equivalent to the “general area” part of a UK postcode (94107 <=> SW15) then the full UK postcode is like the 9 digit US Zip+4 format where the extra 4 digits narrow location down to a block, part of a block or even a specific building.