When creating a Floating Embed with a Google Docs/Sheets template for the first time, you will be asked to sign in to your Google account and authorize access to your Google Drive files.
This is the account you must use to create all other Floating Embeds in the future in all your conversations and cannot be changed; this also means that only the owner of that account can create Floating Embeds using a Google Docs template.
This is a known limitation and will get rectified in a future release of the product.