Can a pyrokineticist's fire lash be made magical so that, for example, it's a +1 shock fire lash?
If so, can a pyrokineticist have multiple fire lashes, each of them different? (For example, can a pyrokineticist have a +1 shock fire lash and a +2 keen fire lash and choose which to use each time the fire lash ability's used?)
The answer is going to depend entirely on how one answers How much like a whip is a fire lash?—at this point, there are two answers to that question, one which reads the various descriptions of the fire lash being a whip as meaning it’s like a whip in all ways, and another which reads the various explicit ways in which the fire whip is described as being like a whip, and infers that these are the only ways in which it is like a whip. Both approaches seem reasonable to me, but neither seems definitively like “the correct answer.”
If the fire lash is like a whip only in the listed ways, magical enhancement is not listed, so it doesn’t work. If the fire lash is like a whip in all ways, then whips can be enhanced and therefore so can the fire lash be.
However, your follow-up question kind of makes things moot: there is no suggestion in Expanded Psionics Handbook that you are summoning the same fire lash every time. To wit:
The whip remains in existence as long as the pyrokineticist holds it,
means that if you stop holding it, it no longer exists—it’s gone, and so are any enhancements you paid to have put on it. Your fire lash isn’t a real, stable object that you’re just calling to your person and then sending back into storage when you’re done with it. It’s a construct of unstable energy that doesn’t exist at all except while you keep it together. You create a new one every time you need another.
So while maybe you can enhance a fire lash, that is a complete waste of time and money since those enhancements are going to disappear as soon as you stop holding it. Maybe if you don’t need to sleep, you can kinda-sorta maintain your grip on it at all times, or you could use a locked gauntlet to make it impossible to let go of, but sooner or later this is going to go poorly for you.
Houseruling the fire lash to allow you to add persistent enhancements to it, fluffed as maybe enhancing you, or your ability to produce a fire lash, seems entirely reasonable to me. Allowing multiple enhancement suites, so you could choose which one to create at a given time, also seems reasonable to me. In fact, both of these houserules would still fall far short of making the pyrokineticist a good class—the class is a pretty pure trap, and I strongly recommend against it.
The issue isn't really with the pyrokineticist maintaining her hold on the fire lash effect indefinitely—adventurers are a kooky and powerful lot, and if an adventurer gets it into her head that she wants to hold something in her hand forevermore, then, by Pelor, she will find a way! The issue is, instead, with adding magical enhancement bonuses and magic and psionic weapon special abilities to any weapon: "Only a masterwork weapon can become a magic [or psionic] weapon," says Creating Magic Weapons (Dungeon Master's Guide 285-6), and a fire lash effect just isn't a masterwork weapon and, for the typical pyrokineticist, never becomes one.
So, until the pyrokineticist can somehow make the fire lash effect masterwork, she's stuck with a regular ol' no-enhancement-bonus, no-magic-or-psionic-weapon-special-abilities fire lash effect.
