Would they be classes, if the base classes didn't exist?
In a global sense, I can't say, because they do. D&D isn't one game out of a field, it's the genre origin.
In a more specific sense, however, where we ask if these new classes would be classes
in a given game where the original classes did not exist, I find that when an MMOG gets "creative" with its class design -- perhaps in order to avoid emulating D&D or WoW, perhaps not -- the results cause me substantial cognitive dissonance.
Here's a great example:
rift.wikia.com said:
The Tactician adds deep and situational versatility to the Rogue calling. Offensively, the Soul combines bolts of withering energy with a devastating set of elemental torrents that erupt in arcs of fire, ice, and death. The Tactician’s group healing skills are just as enviable, as are its fire-and-forget cores and robust support abilities.
...What?
A counter to the Rift phenomenon is Lord of the Rings Online, where the classes are not drawn from D&D, but each represent a single archetype from a low-magic world well. Admittedly, some resemble renamed D&D classes, but for the most part they represent unique concepts with little overlap between primary roles.
So in both the global and specific cases, I would ultimately say no, they would not be classes, because the very name "class" suggests that they are designed to be categorical. To lead a category, a class must simultaneously be specific and inclusive. A fighter fights, a cleric calls upon miracles, a thief steals (in all senses of the word), and a magic-user uses magic. It is difficult to imagine a category that is defined by, say, the Pathfinder Bloodrager, who fights like a fighter, rages like a barbarian, uses magic like a magic-user, and casts spontaneously like a sorcerer. It is highly specific, but also very exclusive. It represents a category of one.
Mashup somethings are a pretty old idea, though, aren't they? Ranger, bard, paladin, those are mashups. The 2e approach to classes was that you are your class, but there are so many options that your class/multiclass can have all kinds of different combinations of things. There are tons of characters with thief skills, cleric spells, weapon specialization, etc.
From my perspective, the bard is only a mashup when executed poorly, and the paladin and ranger should probably not be classes.
I think the AD&D2 approach was excellent, apart from the continuation of racial limitations on class and leveling. Indeed, I have no complaint with multiclassing -- in my opinion, a class should make you feel like your character belongs to a tradition, but multiclassing should make you feel like he breaks from that tradition and becomes his own man.
One of my greatest joys in D&D3 was how much it encouraged multiclassing, and one of my biggest complaints about Pathfinder is how thoroughly it discourages multiclassing.