Well, yes, the code is what distinguishes a paladin from a fighter/cleric variation.
Maybe the 5e design would be cleaner if rangers and paladins were just fighter (or fighter/cleric, or fighter/druid) themes, but I would hope that we don't get too much unwanted complexity from a true paladin class. So long as WotC feels like it can borrow from the fighter and cleric for maneuvers and spells, I wouldn't think that we lose a lot by calling out certain long time archetypes as separate classes.
I'm of two minds about that, and my reservation is that D&D designers always seem to want to meld those two minds instead of recognizing their polar opposite nature. Maybe there is a good way to meld them, consistently, but I keep seeing design problems spring out of the attempt (albeit not always the paladin).
On the one hand, if a paladin is merely a fighter/cleric hybrid, with a bit of character flavor tacked on (the code), then I would expect the fighter and cleric bits to readily stack with fighter or cleric in a multiclass. A character that took half his levels in paladin and half in cleric, I'd expect to be about a 3/4 cleric (or maybe 2/3 cleric, depending on the exact mix). I'd expect the divine spell casting and turning, for example, to be about equal to a cleric of that derived level. That's one viable way of doing a class, as a convenient short-hand for hybrids.
On the other hand, if the paladin is really a distinct class, then I expect the code to do more than be a bit of character flavor. I expect it to be the hinge around which the synergy of fighter and cleric is woven into something notably different than a fighter/cleric multiclass of the same level. That means it has to interact with the fighter-type and cleric-type abilities mechanically in some ways. (Simple example off the top of my head. When a paladin turns undead or demons, it works the same way as it does for a cleric as far as odds, numbers, power, etc. However, the effects are somewhat different. Perhaps the cleric is more focused on causing such creatures to cower and then flee and then be "banished", where the paladin causes them to weaken and eventually turn to dust.)
I don't mind either version. Whichever one is chosen, I'd like it to be consistent across all such hybrid classes, because the choice has a huge effect on how multiclassing can and should work. I don't like a "mushy meld" of the two methods that more or less claims the feel of the latter, but mechanically is the first. You end up with the worst of both worlds, a paladin that is supposedly different, but who if he multiclassed with fighter or cleric would notably weaken himself.
Finally, and separate from the above, I like the idea of a paladin theme, because I like the idea of wizards or rogues or fighters or even clerics taking a vow for such a code later in their career, independent of heavy armor and the like.
