First off, "F/A" is a non-standard designation under the tri-service designation system. "FA" or "AF" would be standard; the second letter would, according to the system, be the "primary mission" of the aircraft, whichever of those the top brass felt the design was better at. The Hornet would probably have been designated "FA" as its primary goal was to replace the aging A-4 Skyraider as well as to supplement the F-14.
The F/A-18 has that designation because the Hornet was originally supposed to be two planes, the F-18 and the A-18, which would differ only in the required avionics and in minor differences to the weapons hardpoints. "F/A-18" became the internal shorthand for the project as a whole. When it was demonstrated that avionics improvements allowed a single airframe to satisfactorily perform both mission profiles, the "F/A" designator was applied to the production design.
The F-15E should, most accurately, be designated AF-15E, adding a "modified mission" designation for ground attack (similar to other modified mission designations such as the RF-4 reconnaissance variant of the Phantom, or the current EA-18G "Growler" variant of the Super Hornet). Its parent design was a pure air superiority fighter which was modified to produce the Strike Eagle, mainly by strengthening the wings, improving engine output, adding CFTs for better range without droptanks, and replacing practically everything under the skin forward of the intakes (avionics, radar, etc). The Strike Eagle retains much of the air superiority prowess of its lineage, and has been tasked with "Combat Air Patrol" missions often in addition to a primary strike mission (along the lines of "go here, drop your bombs, refuel at this tanker and then stay on station for an hour before heading back"), but its primary mission is ground attack (interdiction, deep strikes, SEAD, etc).
The main reason why the mission designation of the Strike Eagle was not changed is that the USAF has rarely made the distinction between an air-to-air fighter and a light air-to-ground strike aircraft. Pre-1962, any one- or two-man aircraft developed for the Air Force was a "fighter" (evolved from the older "Pursuit" designation in use through WWII) regardless of its specific mission profile. After 1962 with the tri-service system in place, if the aircraft is designed with air-to-air engagements in mind, it is a "fighter" in the USAF's eyes no matter what else it does. The A-10 is the sole production aircraft designed for USAF use to be given that designation (which is fitting as it's designed from the ground up to kill armor, and typically carries only a pair of Sidewinder-Ms for basic self-defense); all other "A"-designated designs in the Air Force arsenal since 1962 have been Navy planes first, with the Air Force having to adopt the pre-existing designation under the tri-service rules. As the Navy's never really been interested in any variant of the F-15 since fairly early in the VFX program, it's only to be expected that the Air Force top brass would ignore the attack designation here.