Steven Zahm@ Jorn
re: So is every company a custom post type?
Sorry, no, Connections does not use a CPT for its entries. Rather it uses custom tables with a schema optimized for directories for performance and scalability.
re: Is it possible to integrate it with a membership plugin? So this way the company can claim the listing and add their own info to it.
There is no specific integration with membership plugins, but many users do use Connections in combination with membership plugins in order to a monetize their site. Where access to the directory is one of the membership benefits. There is no “claim a listing” type of feature. You can use Link and, optionally Form, to allow you members to submit and maintain their entry in the directory.
I hope I have answered your questions thoroughly!