You'll have to create an App in each Business Unit (BU) where you need to work with the API for Content Builder. Each of those apps will have their own clientID and clientSecret. You'll have to create a system to generate authentication tokens every hour for each app/BU. Then you'll use the correct Authentication Token for the corresponding BU where you are manipulating Content Builder assets.
Here's the documentation for creating an API Integration Installed Package for each BU, and how to use that to generate Authentication Tokens which you use in the actual REST API calls for Content Builder. This URL has the links that are the most straightforward:
Once you have the Access Token, you'll use it in the headers of your REST calls like this:
Authorization: Bearer YOUR_ACCESS_TOKEN
The Content Builder REST API documentation can be found here:
Keep in mind that anything you're creating in Shared Folders within Content Builder should happen at the top-level (Enterprise) BU.