Yes this is possible by embedding more than 1 Clipchamp button and setting a different upload target per button instance.
This initial setup guide covers the steps necessary to integrate the API.
The way the setup of more than 1 upload destination works is that you need to place the API embed code in the HTML of each page that you'd like to embed a recording & uploading button on.
Next, you need to adapt the upload parameter used for the API on each page to reflect the different upload folder that videos submitted from this particular button instance should be sent to - in the Google Drive example, this would be the gdrive.folder option on https://clipchamp.com/en/documentation/getting-started).
We recommend for a developer of software engineer to perform such an implementation.