That popup should be seizing focus. to the point of suspending the page underneath. As such tab/space navigation should be working as normal there with the user locked out of the page until it is answered.

If it's not, then you've got something REALLY wrong with either the page or the copy of Chrome in question. That's not a "thing".

Is is somehow inheriting something like outline:none from the page (shouldn't as it's a browser control) or is some sort of scripted junk interfering with the browser?

Just ran a text here in both Chrome and Vivaldi forcing that popup, and it works just fine on a vanilla page. Able to tab and space just fine into the allow/block dialogs.

Could the issue be platform specific?