@burnout426 First of all thank you for your help.
Actually I changed the settings in Opera both desktop and Android alike to: DNS over HTTPS via Cloudfare right after installation. I just thought that it was right that, that caused the trouble. But disabling it, didn't help either.
I haven't made a test on another Chromium browser yet. But I remember a time when I had another ISP with some other hardware (router and a Linux laptop in this case) I saw some similar behavior when using Vivaldi. I saw some other people posting similar issues on their forum then.
Actually, my impression is that it doesn't generally happen with Opera but only on specific versions and it seems to me, that it's version 128 on desktop or the respective version on Android that might be causing the trouble.
Yet, I threw it at Gemini and apart from the VPN thing it gave me this answer:
The FritzBox Reaction: Some FritzBox firmware versions have a security feature called "DNS Rebind Protection" or automated DoS (Denial of Service) protection. If Opera attempts to open hundreds of simultaneous connections to a single proxy IP to "speed up" loading, the FritzBox might flag this as suspicious activity and temporarily throttle or drop the connection to protect the network.
2. Synchronization Stress
You mentioned a "fresh install" on Mac, iPad, and Android all at once.
When you sign into an Opera account on three devices simultaneously, they immediately begin a massive sync of bookmarks, history, open tabs, and extensions.
If your FritzBox 6660 is running older firmware, the sudden burst of "Stateful Packet Inspection" (checking every single one of those sync packets) can occasionally cause the router's NAT table to overflow, leading to a hang that requires a reboot.
3. BitTorrent / Flow Integration
Opera has built-in features like Opera Flow and, in some versions, native support for certain file-sharing protocols.
Unlike Firefox, which is very "standard" in its traffic patterns, Opera’s sidebar features and "My Flow" use a specific type of peer-to-peer (P2P) communication to keep devices in sync.
Cable routers (like the 6660) are notoriously sensitive to high numbers of UDP connections typically used by P2P protocols. If the router's processor spikes to 100% trying to manage these, it will keep the Wi-Fi alive but stop routing internet traffic.
I haven't changed the settings in the FritzBox itself yet as it's working for Firefox now. But I may do that in a while.