No, it's not.
When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.
If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.
If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:
You normally interact with an object while doing something else, such
as when you draw a sword as part of an attack. When an object requires
your action for its use, you take the Use an Object action. This
action is also useful when you want to interact with more than one
object on your turn. (PHB 193)
Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.
To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.
By the rules, only objects roughly the same size and weight as actual weapons your character can wield are allowed to be used as improvised weapons :
Improvised Weapons
Sometimes objects not crafted to be weapons nonetheless see use in
combat. Because such objects are not designed for this use, any
creature that uses an improvised weapon in combat is considered to be
nonproficient with it and takes a –4 penalty on attack rolls made with
that object. To determine the size category and appropriate damage for
an improvised weapon, compare its relative size and damage potential
to the weapon list to find a reasonable match. An improvised weapon
scores a threat on a natural roll of 20 and deals double damage on a
critical hit. An improvised thrown weapon has a range increment of 10
feet.
Thus an object too big to be a valid improvised weapon for your character cannot be thrown. Flavor text and the feats' names are sometimes misleading, what matters is what the rules clearly say and what is in the Benefit part of feats. Throw Anything uses "ranged improvised weapons" as key words, it's not valid for absolutely anything your character can lift.
Moreover, the Hurling rage power of barbarians would rapidly become useless if anyone strong enough could throw massive objects.
Best Answer
Quoting from the 5e SRD's section on improvised weapons:
You'll note that improvised weapons are described as "objects". And the closest example to what you're trying to use is "a dead goblin". With that in mind I'd rule that if the enemy is alive and actively resisting you can't use them as a weapon, even if they're grappled, because being grappled doesn't do anything close to making someone "an object".
If for some reason they're not able to resist (i.e. they're dead, they're unconscious, they're paralyzed, etc.) then sure, Enemy A is now an improvised weapon who does 1d4 damage.