You are right, most of the CMS dedicated for organizing image galleries are portfolio type websites for a single user. Joomla on the other hand provides great power and flexibility to built what you are describing.
There are ready made Community extensions like Jom-Social, Community Builder, Easy Social... that will allow you to build a Community driven website. Most of these extensions have integrated features for managing user-groups, photos, videos, discussions, or can be extended with add-ons extensions specific for them.
There are also any other kind of extensions for handling memberships, subscriptions, forums, newsletters, complex forms and more - or extensions that will let you build your Custom Content Types, Directories or Custom applications of any kind inside Joomla (K2, Cobalt, Sobi Pro, Fabrik to name a few).
If something does not already exist, meets your needs, or you just need a specific integration you can always develop it (or hire a developer to do so).
However my main recommendation, if you are new to Joomla and in the web development in general, and you have a clear and specific plan, and your goal is to create a quality online project, you will be better to hire a professional or a team of professionals to build this project on a solid and stable basis from scratch. Unless you have plenty of time and wish to try it at your own while learning from mistakes, you will be better if you make a good research for the right professional (something that one way or another you will probably need in the long run anyway).
The project you want to create is more than the an average one. If you want it successful then you should walk professionally from the beginning to the end.
Update - answer to your comment:
Joomla is well-known for its ability to deliver online communities projects. Overall is a great and stable platform, which incorporates deep ACL support in its core and it has thousands ready-made extensions for any kind of functionality.
Drupal could be also another choice well-suited for big projects, and provides deep customizations flexibility as well.
I am not sure what is the current status of Wordpress community plugins, but at this moment I wouldn't propose WP as the platform for this project.
There are so many other CMS out there, but I do not have that big working experience with most of them.
Finally there is always the choice of a completely custom developed solution.
Keep in mind, that in such projects, it's not the platform that will guarantee its success in all terms (planning, development, launch, operation, marketing-promotion, maintenance, etc), but the guys behind the project.
The right platform at the right hands is a guarantee for success.
The right platform at the wrong hands will be a failure.
A few more things to pay attention when choosing: Community Support, Extensibility, Quality of code base (including system requirements), Existing extensions, Adequate number of developers to hire, Other similar existing projects.
Most possible if I would have to do this project, I would pick up Joomla, mostly because it's the platform I have the biggest experience and know what I can do with it.