This question presupposes that the F-15 has certain unique capabilities that were absent from later fighters. However, insofar as fighters or any other piece of military hardware have truly unique capabilities, they are generally the product of the system as a whole; in other words, to make a fighter with the unique capabilities of the F-15, you essentially have to make the F-15. 

That said, there are many specific capabilities and traits that make the F-15 a good fighter. It started off with a fairly good radar in the initial F-15A models, which has since been upgraded and replaced several times in the F-15C and the -15E strike variant. So, we can say that the upgradeability of the initial radar architecture (including lots of space and power for newer antennas) is one design factor that made and continues to make the F-15 a good fighter. It has two powerful engines which give a good thrust-to-weight ratio (T:W), very important for maneuverability. It was built using advanced manufacturing techniques and materials in order to achieve its strong aerodynamic performance, which of course drives the cost. This gets into perhaps the why of the question. Many other NATO fighters since the F-15's introduction have had good radars or high T:W. The F-15 was developed, in part, in response to fears of new Soviet aircraft. In particular, reconaissance photos of the MiG-25 gave American analysts the impression that it would be a very fast, very maneuverable air superiority fighter. Coupled with concerning performances in Vietnam from contemporary American fighters, the F-15 was developed to meet this apparent threat. Concerns that the F-15 would be too expensive to produce in sufficient numbers led to the Lightweight Fighter Program, which produced the cheaper, lighter single-engine F-16. So, part of it ultimately is that the F-15's capabilities were part of a high-low mix of fighters, and were expensive enough at the time that it was desirable to produce cheaper fighters without all these capabilities. Many of these design traits, like powerful radars, have since been integrated onto later fighters like the F-16 and F/A-18. It is also worth noting that, although it was developed for a different mission, the Navy F-14 Tomcat (a rough contemporary to the F-15) was also a twin-engine fighter with a very powerful radar and high T:W. 

Moving beyond the mid-Cold War period, the advanced fighter proposals of the 1980s actually reflect very strong continuity of design with the F-15. The Advanced Tactical Fighter program that produced the F-22 was in many ways the platonic ideal of the F-15: a respectably fast, highly maneuverable, long-ranged air superiority fighter with a good radar for beyond-visual-range combat. 

As a sort of summary: although the F-15 is the product of a specific institutional and technical context, its key design features are in fact not unique, and show up in several other contemporary and later fighters in varying combination.