Unfortunately, validating clients (whether a Chrome extension, an Android app, an iOS app, client-side JavaScript, or some other client) from a web server is an unsolved problem.
There are some things that you can do to deter abuse and mitigate this problem such as:
- Requiring user authentication (and rate-limiting usage per-user)
- Rate-limiting access on the basis of IP addresses
- Requiring tokens to be provided that are handed out in prior requests (this can be used to ensure that certain APIs are called in certain expected orders / patterns).
- Showing a CAPTCHA or other challenge for anomolous or over-limit usage
While you can additionally check things such as user agent, referrer URL, or a token that you embed in the Chrome extension, with any distributed application, it is easy to reverse-engineer these and mimick them in a counterfeit app, and so these aren't true solutions.