It just so may happen that the characteristics that make the Phoenix rather capable against bombers at super long range, also make it rather capable against fighters at longer than normal ranges.
Namely, Massive kinematic lofting (there are many white papers on this if you want to understand why it helps so very much, including some that relate specifically to DCS) a long burning rocket motor, high speeds in the interim and strong control surfaces in order to maneuver a big bulky missile.
All of this results in a missile that can go a long way, and can maintain its kinematics a long way.
Which means that it can also maneuver when it goes terminal, and keep up with most fighters as well as bombers.
It has a big warhead as well, so it doesn’t have to impact its target (especially a fighter sized target) in order to kill it.
Combine that with the active missile seeker and it really is quite effective against fighters regardless of whether it was initially designed for it or not.
For reference, the AA-10C (R-27ER) weighs nearly 700+ pounds, it is an ENORMEOUS missile (I’ve stood next to one in the aggressor bar at Nellis) the Phoenix wieghs ~1000 Lbs, not a whole lot more. and really is a generation ahead of the R-27ER in many aspects.
Where the Phoenix falls apart is in the maintenance and upkeep side of things, along with the cost of modernizing it. This more than anything else is why it ended up getting replaced by the AMRAAM.