I'm a bit late to this, but a few disconnected thoughts.
Yes, some authorities (and some external contractors) look after other authorities' data on an agency basis, but should use the correct local authority's codes for each of their own / their 'client' council/s.
Some councils (especially the smallest ones) don't want to have to pay for the software to do it. When NAPTAN started, you could upload it to the central data set in Excel (or csv) format and you can't now. Doing that was cheap but potentially fragile.
Following 'austerity', some councils don't really have any public transport staff (or didn't before BSIP funding came along) and I'm aware of some bus operators having difficulty getting NAPTAN numbers out of councils for new stops.
It is possible that mistakes were made on the fringes of council areas when it all started (about 25 years ago now) - some councils contracted out the initial survey to casual staff or a contractor, and it's possible that local knowledge and accuracy was lacking in some cases. Some councils took the task seriously as an important part of providing quality information (the idea of it feeding real time data and many people having apps on a device that does GPS location was probably less of a concern then. I remember going out surveying bus stops with a rucksack full of electronics to plot the locations for my then council, which was probably less accurate than what a basic smartphone can do now. And it didn't work at all in the 'ground floor of a shopping centre' type bus station!) Some saw it as a 'burden' placed upon them by central government and did it reluctantly and as cheaply as possible.
The guidance at the time also included the concept of having a 'pair' or 'cluster' of stops - there are locations where each stop in a pair (sometimes the stops on opposite sides of the road) are in different authorities. Some authorities took the administrative location as the highest priority, so you get the bus stop on one side of the road with one locality and stop name, the opposite stop can have a different locality and stop name. Other councils talked to each other and agreed that council A would own this pair of stops, council B that pair (or something like that.) There must be places with a 3 or 4 way boundary and bus stops round a crossroads in different local authority areas.
In some cases, it may be that a bus route that is almost entirely in authority area A ventures across a boundary for one stop, and all concerned agreed that council A would take ownership of physical infrastructure and / or NAPTAN at that one stop.
Administrative boundaries can move slightly for one reason or another - it's not unknown for one property to be split between two local authorities (one of the schools i went to, a borough boundary ran through the middle of the site, but it's now wholly in one London borough - I presume it didn't matter when ILEA was still a thing) but these have tended to be 'tidied up' in recent years, it must make rates / council tax complicated for a start.
Alternatively, it's not unknown for a bus stop to get moved slightly for one reason or another (e.g minor road layout change, new lamp posts) and a few stops could have moved in to a different council's area and nobody has noticed - councils don't have the resources to go out and re-survey, and in some areas, ownership / maintenance of bus stop infrastructure is with bus operators not councils. (In Berkshire, there were a few bus stops still showing Tillingbourne ownership and timetables until only a few years ago, on a section of route that wasn't reinstated after Tillingbourne ceased trading in 2001 - I think they have either been taken down, fallen down or been lost in the undergrowth by now.)
Then there are the local government reorganisations which have split or combined existing transport authorities, and it's again not impossible that some mistakes have happened in re-allocating bus stops at this stage.
I'm not sure if there is a process for 'moving' an individual bus stop or two from one authority's data set to another - in theory council A could set up 'new' stops at a location and council B could make the same stops 'deleted' in their own data, and get the bus operators to change the stop number they use in systems at an agreed time.
(And yes, I have been involved in this in one way or another for much of the last 25 years, I'd sooner not say where, though...)