[00N8]

IMO both usages are correct, although yours is probably more common. 'Dubious' here means something like questionable/uncertain & it can apply either way: The distinction can be 'questionable' in terms of whether it's something to be proud of, or in terms of whether it's even true in the first place.

[quesera]

The phrase "dubious distinction" only means one thing though.

If you parse the words separately, the other meaning is literally correct, but will not match the speaker's intent.

[FabHK]

I had a similar discussion here with someone who thought that "dynamic programming" was the act of programming in some sort of dynamic manner. Not sure I managed to disabuse them of that notion.