Update 2020
I’m not sure exactly when, but Github has (somewhat recently) added the concept of projects, which kind of fill the missing gap. I would argue they aren’t quite the same as Bitbucket Projects but they are better suited to grouping related repo’s in Github than Orgs
Original answer
Organisations in my opinion fit a different purpose in Github than grouping repos (although they do serve to group repos). Organisations are more about fine grained control around repo access (thats my understanding).
Bitbucket has introduced the concept of “Projects”, with the following hierarchy (with a comparison to Github):
Bitbucket: Team -> has N -> Projects -> has N -> Repos
Github: Organisation -> has N -> Repos
Bitbucket still allows Repos to not be assigned to an team or project, I am guessing to support older repos that existed before the concept of a project.
To answer the question, no, not directly. There are outstanding requests with Github to add groups, but it doesn’t seem likely (at this point in time).
Prefixing works as a so-so solution:
Repo name: [project]__[repo name]
Lets say you have a client “acme” with two repos:
Eg: acme__api
Eg: acme__landing
Github’s search is quick and inline, so doing a search for acme__ in your repo list will list all repos for the acme__ project.