Webster's II (1934) gives dog collar in that sense as Brit slang. Someone with an OED could say how far back it goes, and whether or not it’s usually derogatory.
‘Maurice, why did you bring that ghastly little dog-collared drip over here like that?’— Kingsley Amis, The Green Man, 1969.
“Dog Collars”
This is not a derogatory usage, nor is it recent. It has been the commonly used term for a clerical collar (all denominations) in Britain for more than a century. The first recorded use in the OED dates from 1883.
It's the Anglican collar as distinct from the Roman collar (see above).
I love "ghastly little dog-collared drip," and I'm saving it for future use!