That's not correct.

A "dubious distinction" is a distinction which is undesirable.

West Virginia has the dubious distinction of being the US state with the lowest average educational achievement.

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.

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.

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.