I have ID'd 2 family members in my life and I have also been on the scene with the coroner to remove the body, twice . The morgue will usually call next of kin, family member, etc. Not sure about them taking "requests". But then again, if a "stranger" has found u, then I guess they will...
However, if someone (fam member, friend) finds u n has ID'd you when the coroner gets there (filled out forms), then they will take that as the ID'ing of ur body n u will not b able to request someone else..
(I'm in NY so could b diff from UK..)