If you don't use Safari the OS will nag you to use Safari until you launch it whenever you open a non-Safari browser. You can temporarily ignore it for a bit but it will come back. And this resets every time Safari is updated. They will also stick a notification badge on system preferences if you're not using iCloud.
I've never had that problem on any machine or OS version.
I do have a pointless notification telling me there's a new version of macOS I could update to, and the reason it's pointless is that my mac is too old for the specific new version of macOS that the notification is trying to convince me to install.
But that thing you're getting with the web browser, I've never had that problem.
My personal mac is old, but the OS is newer than that link.
My current work mac (and all previous work macs back to when my current personal mac was new) have never had the bug you're talking about — install chrome, make default, never get nagged.
I haven't had this problem ever, and that goes back to when Safari was first released. Yes, on a new machine, first launch of another browser... after that I click the button to make that the default browser and then never happens again. And I've gone through a LOT of machines since Safari was released.
I use MacOS Monterey. Never been nagged by OS to use safari instead of my current browser. Safari does have a "set me as default" nag I cant disable, the few times I've launched it - but this is in the browser window, not a notification.
Never logged into iCloud and have never seen this notification badge.
