VOR and ILS frequencies are paired with specific DME frequencies, so it naturally follows that if you have a VOR or ILS frequency selected, you will get the appropriate DME station (and audio).
I wouldn't bet on that.
Its not as though there is a mechanical connection, its two separate avionics boxes being signalled by the control head on the flight deck. They can (and do) glitch from time to time. I've had a VOR box hang on a previous freq whilst the DME happily moved on. This would have left us intercepting a VOR radial instead of the ILS centreline, and the VOR was a couple of miles to the side of the airport. Early (non) confirmation of ident made this a non event.
More broadly, people should be in the habit of identifying all aids, DME included. e.g. the near CFIT by a BAe146 following the wrong DME arc.
People get blase about this, because you can go for years and years getting exactly what you think you've dialled up.
pb