I had to look this up to confirm. I can't find anything in 5e that says you can military press this weight, just that it's the threshold for encumbrance. Unless you assume that "lift" means "military press".
Player's Handbook said:
Lifting and Carrying
Your Strength score determines the amount of weight you can bear. The following terms define what you can lift or carry.Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don’t usually have to worry about it.
Push, Drag, or Lift.
You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.
Size and Strength.
Larger creatures can bear more weight, whereas Tiny creatures can carry less. For each size category above Medium, double the creature’s carrying capacity and the amount it can push, drag, or lift. For a Tiny creature, halve these weights.
[MENTION=6801328]Elfcrusher[/MENTION] - Edit: Sorry, you edited your post. But the rules are vague about lifting. Also long as it is under your carrying capacity, the rules are (I suspect) purposely vague. But theoretically, a character could lift their carrying capacity without problem over their head or any penalty to movement.
In Game of Thrones (Fantasy Genre) we see the giant Wun Wun
-Smash a dude against a brick wall
-Knock out a horse and send it flying 10ft back with a single punch
-Casually grab and toss a phalanx soldier 20ft+ into the air.
Mechanically, Wun Wun could be described as a Goliath Barbarian under the effects of an Enlarge spell. If we assume an 18 Strength (easily achievable by level 4 with Standard Array), this would make the player capable of carrying of 1,080. Push, Drag, and Lift is 2,160. If we assume that the Goliath also took a potion of Hill Giant Strength (Uncommon potion, increases strength to 21), then carrying capacity is 1,260 lbs and Push, Drag, Lift is 2,520.
For perspective, median height of a Goliath is 7'1". According to Volo's, this would put the Goliath in a median weight of 277 lbs. However, real world comparisons of strongmen (what seem to be the closest real world analogs to barbarians) of similar height to a goliath mean a more realistic weight would be in the range of 350 - 400 lbs. Sean O'Hagan is 6'11" weighing 440 lbs, Hafthor Bjornsson is 6'9" and weighs between 397 - 441). A Goliath under the effects of an enlarge spell would 14'2" tall and weigh between 2,800 - 3,200 lbs.
An Olympic Shot Putter can throw a 16 lbs metal sphere about 60'. The hammer throw (16 lbs sphere on a chain) is around 260'. If a common weight for these athletes is around 250 lbs, this is an object weight about 6% of their body weight. For an enlaged Goliath Barbarian, 6% of their body weight ranges from 168 lbs - 192 lbs. Assuming equivalent strength ratios, an Enlarged Goliath Barbarian can throw an average sized human nearly as far as Olympic athletes can throw a shot put or a hammer. Of course, a shot put and hammer are perfect instruments for throwing and thus ideal for that purpose. However, even if we half those distances, an Enlarged Goliath Barbarian could throw a full grown man 30ft, or 130' if there is a harness that the player could grab onto.
So what I'm saying is that while not just anyone could throw a person in D&D, it is not outside the realm of possibility either, nor does it require a Genre change to imagine.
