The “fighter” designator is applied to many tactical aircraft because fighter pilots have long ruled the roost in air services. It’s mostly about salving some of the biggest egos around.
This ties in with my second point: there are very, very few single-mission tactical aircraft anymore. They started disappearing in the ’60s with the F-4 and F-111. The trend has continued over time, and it’s a rare aircraft and air force that can afford to specialize on a single mission. To that end, the “F” designator is as useful to describe a tactical aircraft as anything else. Think of “fighter” as in “fighting”, and you’re there. If your plane can carry air-to-ground and air-to-air munitions on the same mission, it’s a multipurpose aircraft.
That brings in my final point: close air support is just a mission, not a platform. The whole idea of a specialized “CAS” designation that applied only to the A-10 smacks of an odd way to try to fight the battle over the future of that particular airframe.