The wikipedia -onym article presents dozens of name-related words ending with onym, none of which exactly apply to your question. (An eponym, for example, is "the name of a person or thing, whether real or fictitious, after which a particular place, tribe, era, discovery, or other item is named or thought to be named", while a demonym is "a name for a resident of a locality ... [frequently] derived from the name of the locality". Unfortunately icononym is not listed in the article.)
A hypernym of the word you seek is taxonym, "a name used for classification or identification purposes", but for better specificity the term groupname seems appropriate. For example, in
1912-1925 it was used in sentences like the following, where its meaning is name of a group:
Judging from the reports of ... expeditions, they must have heard while there the native groupname Hasinai;