I hope it is clear to everyone that there is a difference between the Group’s definition name (which is assigned sequentially as Groups are created and cannot be edited) and a Group instance’s name (which is today assigned afterward via Entity Info and is then displayed in the Outliner and Entity Info).
The only benefit I can see to displaying the sequential Group definition names in the Outliner or Entity Info is to let you see that you have (or have not) made copies of Groups. Other than that, the sequence numbers have no useful semantic value. Quick, was Group#134 the railing, or was that Group#135?
On the other hand, though, it seems that displaying them would do no significant harm vs the existing ambiguously uniform “Group” in Outliner or complete absence in Entity Info.
I agree with @TIG that if you want to name Group instances there is an existing mechanism so a new preference and dialog aren’t necessary. I also agree that there are various differences between Groups and Components that are mainly historical since Groups have for some time been implemented as a special case of Components. Axes are a good example. These seemingly arbitrary variations have no basis in the real differences between Groups and Components and are a source of confusion to newbies.
There are also aspects that should stay different, as they serve a valuable purpose. The exclusion of Groups from the Component window, with corresponding elimination of thumbnails reduces file size and gives a tremendous performance boost to extensions that generate a lot of Groups .