I had the same problem. I'm not sure if all the following steps are necessary but here is what I did to get it working for Windows 10 (64 bit), Opera 49.0.2725.64:

1. Uncheck the Setting "Offer to save passwords I enter on the web" (found in Settings | Privacy & security)
2. Close the browser and re-open it.
3. Check the Setting "Offer to save passwords I enter on the web"
4. Refresh Settings Tab
5. Refresh the Site/Page for which you want to store the password. Enter the username and password.

Now you should see the prompt to store the password in Opera's password manager.

I had the same problem. I'm not sure if all the following steps are necessary but here is what I did to get it working for Windows 10 (64 bit), Opera 49.0.2725.64:

1. Uncheck the Setting "Offer to save passwords I enter on the web" (found in Settings | Privacy & security)
2. Close the browser and re-open it.
3. Check the Setting "Offer to save passwords I enter on the web"
4. Refresh Settings Tab
5. Refresh the Site/Page for which you want to store the password. Enter the username and password.

Now you should see the prompt to store the password in Opera's password manager.