A lot of players can get CDDB info about Audio CDs, but I have never seen that in Clementine.
Given the prestige and other qualities of this player It's hard to believe that it lacks this feature.
The closing comments of Issue #314 indicate that Clementine uses the
libtunepimp library for talking to the MusicBrainz service for this tagging capability. Unfortunately, per the MusicBrainz page,
libtunepimp uses their deprecated web service (see http://wiki.musicbrainz.org/Web_Service for more details). Which means that Clementine may not be getting all of the latest-and-greatest from MusicBrainz.
Now, Issue #314 is more about get automatic song tagging support in general into Clementine, not necessarily CDDB specifically. And indeed, among the open Clementine issues, there are a couple which suggest that CDDB support is lacking/not working as expected (e.g. Issue#3067, Issue#4120).
So I think that the answer to your question of whether Clementine lacks CDDB support is most likely: "Yes, it's CDDB support is lacking enough to be considered missing". Sadly.