The Monk can make a long jump up to 32 feet.
As you quoted,
Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump.
If the monk's Strength score is 16, they would normally be able to make a long jump of 16 feet with a 10 foot running start. Thanks to Step of the Wind, that distance is doubled to 32 feet.
Alternatively,
The Monk can make a high jump of up to 12 feet.
The rules for high jumping state:
High Jump. When you make a high jump, you leap into the air a number of feet equal to 3 + your Strength modifier if you move at least 10 feet on foot immediately before the jump. When you make a standing high jump, you can jump only half that distance.
With a Strength of 16, the monk has a +3 modifier and can usually make a high jump of 6 feet with a 10 foot running start. With Step of the Wind, this is doubled to 12 feet.
The Path of the Totem Warrior subclass gives specific abilities at 3rd, 6th and 14th level specifically as outlined in the subclass description.
For your player, who is using the Tiger totem spirit (but for their own purposes, representing it as a Lion), this would lead to them gaining:
3rd level (Totem Spirit - Tiger): While raging, you can add 10 feet to your long jump distance and 3 feet to your high jump distance.
The spirit of the tiger empowers your leaps
Note that, at level 6 and level 14 you can choose a different totem animal. You do not have to commit to your choice you made at third level.
6th level (Aspect of the Beast - Tiger): You gain proficiency in two skills from the following list: Athletics, Acrobatics, Stealth and
Survival.
14th level (Totemic Attunement - Tiger): While you’re raging, if you move at least 20 feet in a straight line toward a Large or smaller
target right before making a melee weapon attack against it, you can
use a bonus action to make an additional melee weapon attack against
it.
The only other thing the physical totem can do is listed in the description of the Totem Spirit section, emphasis mine.
At 3rd level, when you adopt this path, you choose a totem spirit and gain its feature. You must make or acquire a physical totem object — an amulet or similar adornment — that incorporates fur or feathers, claws, teeth, or bones of the totem animal. At your option, you also gain minor physical attributes that are reminiscent of your totem spirit. For example, if you have a bear totem spirit, you might be unusually hairy and thick-skinned, or if your totem is the eagle, your eyes turn bright yellow.
But, it seems like you understand all that already.
What you have is a problem player. You have identified that the player is, at best, misreading and misunderstanding the nature of the subclass, but you have also identified that the player is blatantly cheating by using the benefits of the other Totem Spirit animals.
Depending on your situation, and your group dynamic, it's up to you how you want to handle it. I assume you're the DM in this game, so generally the pieces of advice to dealing with a player that doesn't want to follow the rules/plays in bad faith are:
Explain why the rules are there in the first place.
The game is boring for the other players if their characters feel comparatively weak and ineffective.
D&D is about fun. Fighting about rules isn't fun.
This is largely self-explanatory, but try to have the player understand why you dislike having these disagreements. The game is not designed to be adversarial in nature.
Don't play with this player.
If all else fails, there's no reason to spend time playing D&D with a player that openly cheats and uses bad faith arguments to cause division. Find a new group.
Best Answer
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:
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.