Amber is not a browser extension. It is primarily used for native applications, such as Amethyst, that will automatically reach out to it for login, if they support it.

For browser login for web apps like zap.cooking, you would need to install a browser extension such as Alby or nos2x, depending on what is available for your web browser of choice.

Since you are using Amber,I assume you are on Android. I would recommend either Firefox web browser or Kiwi browser, and pair them with nos2x, which will store your private key/nsec.

Alternatively you can install Spring browser from zap.store, which is a web browser designed specifically for nostr web apps, and it has a signer extension built into it, along with the ability to connect any Nostr Wallet Connect enabled wallet.