(Client area?) Allow users to only edit/see their own posts

This is already possible yes with the “Author” role provided by the v3.1.x release feature called RBAC (Role based access control). Authors will only be able to see/edit their own content

This level of control requires the EE version of the RBAC feature, or something like this you will probably want to use the user REST/GraphQL routes in your frontend as there is no limits and total customization ability of those. The Admin panel is slightly limited in control given to the CE users.