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).
The ident you get on the lower part of the ND is VOR/ADF only with VOR/DME readout, according to the FCOM.
The reason I ask, is that on previous aircraft I have flown, the ASP would have separate DME selectors.