Is "Menu -> settings -> privacy and security -> passwords -> offer to save passwords" checked?
Do you have any extensions? Maybe one of them is messing with things. Do you have Opera's ad blocker on? Does this happen with all sites you've tried or just some? Are you using Opera's VPN or Opera Turbo?
Download the Opera installer, launch it, click "options" set "install path" to a folder on your desktop, set "install for" to "standalone installation" and install. Does it at least work fine in that Opera?