The concept of 3e prestige classes was built around requirements. You had to complete the mechanical requirements and (possibly) also some narrative requirements in order to enter a prestige class. Then, the prestige class worked like a regular class in the sense that it granted you a progression.
This makes it hard in 5e to use subclasses for the same purpose, because subclasses have a fixed level entry point (usually 2nd or 3rd level), which is actually very early on. How can you become a member of a "prestige" group so early? You have to get your subclass at that level, miss it, and your choice is over, unless you change how subclasses work (there's definitely a chance the Advanced game let you mix subclasses tho).
Furthermore, subclasses have no requirements, and it would be pretty hard to design a reasonably interesting set of requirements when you only have 1-2 levels to make sure you get them! (note that in those 2 levels you almost only choose skills and 1st level spells, but no feats and no higher level spells).
Because of these, feats (or feat chains, when one is too few) are actually a better system to represent 3e prestige classes in 5e. At least, you get multiple entry points at different levels (although not every level), thus you can define requirements better (including the minimum level if wanted), you don't have to tie them to specific classes, and you can design a whole feat chain to implement a progression into the prestige class.
But ultimately, for an even better representation of 3e prestige classes, 5e would need a new system, which can simply be... 3e-style prestige classes! It should not be hard at all to add to 5e, and with the new multiclassing rules about spellcasting, it should work much much better than in 3e. However, I don't see this coming until maybe a few years from now.