In the NG the identification is done automatically and boeing does not see the need to identify navaids via the ASP anymore. It happily identifies DME-only stations as well, not only VORDME or ILSDME ones. Yes, it can happen that it shows a different identification than the DME used, most common when there are two VORDMEs using the same frequency within reception range (TGL and MUN for example).
On the classic you can identify the DME by waiting until you get the morse code on a higher pitch, but you have to plan some extra time for it as it is only every 30 seconds or so.