Microsoft Word and Microsoft Excel have the ability to open documents by URI (for instance pointing to a web server) and to save them back after they got edited. This functionality can also be invoked by a browser. This, for instance, is done in products like Atlassian Confluence and Microsoft Sharepoint.
Is it possible to open an Adobe InDesign (while InDesign is installed on the computer, ofc) document in a (similar) way like described above?
This means:
The short answer is no.
You can build a system with web front-end and InDesign Server as a back-end. This will achieve the functionality you describe and user will not need a local copy of InDesign.
Another, simpler(and cheaper) option is to use one of the cloud services, like dropbox, or Adobe Creative Cloud with a shared folder and share the files for collaboration.
May be, there is a plugin capable of doing something similar, but I did not come across anything like this. So, another option is to build a plugin, although it will have to be distributed to every end-user and you will need your web service to server the files.