launcher --show-component-extension-options to see the extension in opera://extensions. Then, click "details" on the extension and look in the address field or turn of developer mode before clicking "details" and look at the details, and you'll see the id, which is "ionkhgehfolinkdpgdbinmgbfaonpcnk".
You can look in Preferences, Local State and Secure Preferences to see what you can find after clicking "Not now" and closing Opera. While I did find mention of that ID, I didn't see a setting anywhere that looked like a "user said not now" setting or a "last timestamp user said not now" setting. But, maybe you can find it.
I did ctrl + shift + i for developer tools on "amazon.com", switched to the "Applications" tab and cleared different storages and cookies etc., but none of that cause the extension to pop up and ask to be installed when visiting amazon.com again. So, I'm doubting it's store there.
I looked at a few other files in the profile folder, but didn't see what I was looking for.
An alternative might be to install the extensions and then disable it in opera://extensions. Maybe it'll stop bugging then since it's already installed.