I was actually about to start a thread of my own to ask exactly this!
“I need to make an appointment with doctor.”
I’ve lived in the UK my whole life and have never heard anybody use ‘doctor’ in that way. I would say, ‘I need to make an appointment with the doctor (if there is only one doctor at the practice and I don’t know his/her name)/a doctor (if there are two or more doctors at the practice and I don’t know their names)/Dr Smith (if I know the name of a specific doctor).’ I would never just say, ‘with doctor’. ‘Doctor’ is an academic title and job, not a name.
I’ve never seen people do this anywhere but CAF and it always sounds weird to me.
Same here. I have never heard anybody say this. I am surprised to hear that it may be a non-American thing, as I assumed that the explanation was that most people on CAF are American and guessed that it must be an American thing!
It just doesn’t make sense to me. The only person I would refer to as ‘Father’ would, just possibly, be my own father, and only then if I am talking to a close family member. With regard to priests I would address a priest as ‘Father’ if I don’t know his name or as ‘Father+Name’ if I do. When referring to a priest in the third person I would say ‘Father+Name’ or would say ‘the priest’ (or more specific job title, e.g. ‘the cathedral administrator’). I think it just sounds strange to say, ‘Father gave a good sermon today.’
I’ve even heard the plural: “the Fathers at Paris X are great.” Or “one of the Fathers will come to hear confession.”
That is different, as ‘fathers’ in that context (capital letter not required) is using ‘father’ to denote the clerical office or, quite simply, job description. Hence it is used with a definite article. One can also, less commonly, use it, with a definite or indefinite article, in the singular: ‘I went to the church and spoke to the father’, ‘My uncle is now a father at Ampleforth’, etc. That is using ‘father’ as a synonym for ‘priest’, whereas what the OP is talking about is essentially using ‘Father’ to replace the priest’s name.
In short, I find it strange, and I’m not going to begin doing it myself. ‘Father’ is not the priest’s name, and it seems strange to refer to him as ‘Father’. I’ll continue to say ‘Father John’, ‘Father Smith’, ‘Good morning, Father’, etc., but I’m not going to start saying, ‘Father prefers to wear black vestments for funerals’, or, ‘I must ask Father about next Sunday’s hymns.’