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 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.