group in the context of the indieweb (also "indie group" or "indie groups") is a place where people can deliberately share content with each other, not necessarily on their own domain (though likely copied from via webmention etc.).
Hashtags are similar in that they're a mechanism for deliberately sharing content with an explicit focus (group-like).
incidental vs other place
However hashtags are not a "group" in that you're not going some other place to explicitly scope your post primarily to that place. With hashtags you're only incidentally (typically with merely adding a # to your existing text) associating your content with others.
your own publishing vs group sharing
Hashtags are about publishing your stuff first, and associating it with a set (like a group) second. Groups and posting to groups are about primarily about sharing with the group first (though if it's a public group, you may also be secondarily publishing what you share publicly).
unmoderated vs admin
Groups usually have "admins" of some sort (typically starting with whoever created the group) that have the ability to block, include, exclude, remove content.
Hashtags are usually completely unmoderated - anyone can post "to" them.
Related IndieWebCamp Sessions
and extract from that, add observations, examples, here.
Private Groups To Reduce Facebook Usage
- Tantek Çelik has a brainstorm documenting use-cases and some design thoughts for Private Groups so he can reduce his use of Facebook private groups.