I need to develop an application in which the data is recorded as a group owner, not user. A group may have multiple users, but also a user belong to multiple groups, although representing only one at a time.

It is possible? How?

Yes, it is possible for users to belong to many groups, and for a group to contain many users. For information about creating groups, take a look at the user guide (http://devcenter.kinvey.com/rest/guides/users#usergroups), and for information about how to set an entity's ACL to give read and/or write access to a group, take a look at the security guide (http://devcenter.kinvey.com/guides/security#entityanduserpermissions).

It is necessary for a user to keep information shared with users of their own group.