It certainly does. I'm pretty sure that almost every IRC client can do this, as can IM clients like Pidgin. (That reminds me: does Hipchat work well with Pidgin?)

edit: I think the difference on that chart, though, is that mIRC is a __client__, whereas I suspect that Hipchat's server also does logging -- so that you can see the chat log even if you were previously not connected. That's not something that is normally suported with IRC. You could roll your own with a custom client and server, I expect, but at that point you'd be investing probably more effort than it's worth.