What I am about to describe is common but by no means universal, and before I do write, would like to comment that as a priest, I am considerably blessed by the reciprocal and open relationships I share with many of the Funeral Directors with whom I work.
The simple fact is that I have done lots of funerals services and value its ministry considerably (as do the families I have served). It has also given me a significant insight to the world of Funeral Directors and pressures that they face in meeting the acute needs of their clients.
In an ideal world, a client would seek out a FD, and as part of the transaction of arranging the funeral, may disclose a need in faith. That is, more often than not, a denominational identity ("mum is/we are Catholics/CofE/Methodist, and so on). Once known, the understanding is that the FD would contact the nearest minister of that denomination within whose pastoral boundaries the deceased lived and commission their services to lead the funeral. That means that if mum was a Catholic she will be given her send-off by the RC priest in the area; by the local Vicar is she was Church of England (also the default if their faith is known but not the denomination); or by a anti-faith activist if mum claimed humanism among her character traits.
I have sympathy for the FDs here. Before them sits a grieving family and they are keen to meet their needs for their sake and indeed for the sake of the business for whom they work. They may fail to reach the Catholic priest or the local Vicar in the moment that they sit with the family, so what do they do? Do they wait? Do they inconvenience the family by delaying a service date for their dear departed mum?
Very often, and for quite understandable reasons, they ring the friendly ash-cash collar-for-hire who is their pet minister and often has no "parish" so is easily and immediately contactable. The date is booked, their pet minister will look after the the family, job done.
If your job is within the realm of "getting the job done" by the tick-list - then this would be fine. Then imagine that dear departed mum was the pillar of local parish church. In that situation, often the family will know and will be specific about the priest, not the denomination, and the job is arranged. More and more often though, people's faith habits become eclipsed and so they may know that mum was a CofE person but not a lot more. The FD tries to ring the Vicar who is, for sake of argument, running his parish and not able to take the call at that moment, and so it is that the ash-cash-collar is commissioned, the date booked and everyone is happy. Mum gets a 'vicar' of some sort, the family leave happy.
Except that the ash-cash-collar is a, for sake of illustration, a Methodist or other. You may imagine that it makes no difference, but to a lifelong Anglican who had a chance throughout their life to change denomination, it does make a difference. A big difference. The service is done, it's nice, boxes are ticked. But there may be another factor. In many cases, the deceased wasn't a current congregant often through infirmity and may have moved into a care-home many years before the present Vicar arrived. The direct connection can be, and often is, fractured so that the death itself isn't known at parish level.
What happens to the rightful pastoral needs of the family in this instance? Where is the follow-up? The amount of times I have visited the Crem to see a Church of England person being given their once-only send-off by, for example, a Methodist minister - you'd be astounded. And I doubt that the pastoral needs of the family are met because to ash-cash-collars, this is big business (a figure of 40k per annum was mentioned in a chat I had with someone yesterday). The funeral is not the end of the process - it is often the beginning. The follow-up is of the highest importance and is quite often therefore non-existent for a significant number of families who took the minister who answered their mobile first. As I have sympathies for the FD who have a job to do, so I have greater sympathy for a family whose mother simply got the wrong funeral, perhaps in a theological language that they would never have understood or subscribed to, in the presence of a man or woman who wouldn't have ministered to them in the first place.
I can't pretend to know what the answer is, but it is a big issue certainly in the minds of many Anglicans (whose opinions I hear most) who have simply given up trying to minister to their dispersed flock because they barely get a look in through the friendly arrangements put in place before they themselves moved into the parish.
I sat with all the Christians ministers in my area yesterday for lunch. I mentioned a name to them of a chap who does much in the town in this ministry - and not one of us knew his name or who he was, including the person who shared his denomination. She hasn't done many funerals at all - and now I think she knows why. I doubt, too, that those who claim her denomination in the area are immortal, either.