The customer ID must be included in the request, so the provider knows that the request came from you, as @Matt Lacey already indicated. Typically, the provider also provides the client’s secret secret, which is additionally included in the access token request, so the provider can check if your application is allowed to use this client identifier.
Chrome extensions run on an open platform, and the platform itself does not provide any methods for authenticating an extension on a server (which would also have to support an additional resource) or preserving properties (it would be difficult, if not impossible, on an open platform), so maintaining privacy client confidentially, unfortunately, is not possible.
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