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.
It reduces the effects of battlefield control on the Barbarian.
The benefits are minor, but can be significant in the right situation. The primary ability of the Barbarian is Rage, which among other things:
When you make a melee weapon attack using Strength, you gain a bonus
to the damage roll that increases as you gain levels as a barbarian.
Thus, the classic barbarian build utilizes melee attacks. They are not usually envisioned as ranged attackers. True, they can throw a weapon, but then they have lost their attack. They can also use missile weapons, but then they lose some of the power of their main class feature.
This makes barbarians easy to nerf by an opponent with some degree of battlefield control. Shooting arrows at them from a cliff top or the other side of a chasm, or flying just above their head, or any other way they can deliver damage to the barbarian from a vantage point of remaining out of melee weapon range.
The Jumping ability of the Tiger Totem can reduce this weakness by giving the barbarian some recourse toward reducing their disadvantageous position. Now they can jump up that cliff, over that chasm or just up to get the person buzzing over their heads. It's true that the enemy could still be too high or far to reach (the extra jump isn't that much), but the Tiger Totem jump does seek to reduce the number of circumstances where the barbarian cannot respond due to distance.
Best Answer
Per the rules for jumping in the Adventuring chapter (Basic Rules p64, probably the same section you cite):
Yes, you can jump in combat as part of your movement; no, you can't jump further than you could move anyway. Rules as written, the Jump spell doesn't extend this distance.
If I were the GM, I would say that the Jump spell makes every three feet you clear cost a single foot of movement, but that would be a house ruling.