my + noun

[burro]

Hoey has pointed out that "while it is quite usual to speak of my doctor or my accountant, it is not usual to speak of my carpenter or my
bricklayer", but isn't this just because we see the first two more regulary, or we don't change them as often as we would a carpenter, for example?

I would guess the Lord at the manor would say "my carpenter/gardener/etc.".

[Anglika]

I think this will depend on context. There are two in which I can see "my" being attached to a workman in this way - as you say, an aristocrat with a large staff of workers, or if you are having work done in the house:

My electrician has had to rewire the entire building.

If you want to come round and I'm not there, my carpenter will let you in.

It is a colloquial use that is not uncommon here. It merely means someone who you have employed.