Not meaningfully, no. A UK postcode covers 20 households or less. If you have that plus gender and date of birth (as seems to be the case here), you almost always have a unique individual.
NHS England’s Chief Data Officer says that with the pseudonymous dataset (he calls it amber data) patient’s identifiers are removed, including "(their date of birth, postcode, and so on)". Also, I think it might be hospital number rather than NHS number in the data
But, I can see why all the other data are statistically relevant in analysis.
Postcode, so you can track viral outbreak etc with a fair degree of locational accuracy.
Date of birth, so you could analyse if people born in certain seasons had higher incidences of various diseases. (You could probably drop the day and just have month, but there could be cases where there were more hospital deaths for babies born on a Monday, etc).
A hash of the NHS number would give a UID without compromising the person's details. Providing a UID that is one-way linked would allow HSCIC to go to the actual data if a care situation warranted it.
I'd argue that the first half of the postcode and month of birth are ample for outside-HSCIC use. Monday-deaths is the kind of thing that's done internally anyway.
Tiny nitpick[0]: a UK postcode corresponds to a minimum of 1, and maximum of 100 (not 20), delivery points[1] (not households). Typically about 15 delivery points.
I fully agree with your main point, though, that the data are not meaningfully anonymised.
Not meaningfully, no. A UK postcode covers 20 households or less. If you have that plus gender and date of birth (as seems to be the case here), you almost always have a unique individual.