A WoW warrior class is a tank. It can give and receive a dishing. A DnD rogue class is not a tank. It would not move forward into battle in a frontal assault, with the intent of pushing a breakthrough.
Being able to take unreal amounts of damage is sort of general to video games, and not specific to tank character classes. The medic in TF2 can survive more direct hits from a rocket-propelled grenade than a real-world battlefield medic, too.
So the point of a "tank" is to run in front of the party. If the enemy tries to run around the tank, they get "CC'd" (crowd-controlled" and are then unable to move. The DPS-job is then to run in, kill them while they're under the control of the tank (aka: while the enemy can't move), and that's that.
So trying to kill the enemy DPS often just gets your own party out-of-position. As such, you're often forced to kill the enemy tank first.
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Needless to say, these fantasy fights / video-game fights don't exist in the real world. These roles were setup by video games, probably World of Warcraft and Diablo II before LoL and DotA. (Maybe an earlier game than that?)
But when video gamers hear the word "tank", they expect real-world tanks to act... well... like a video-game tank, and not like a real-world tank.
With PVM, there is no focus from the monsters directly, they have aggression algorithms, and tanks typically have a way of directly manipulating this (draw aggro). They have skills that improve statistics usually defensive, and sometimes can outright prevent damage with well timed skills. They're "Tanks" because they absorb/prevent damage while keeping attention on themselves.
In PVP, Tanks are typically fight starters or stoppers, that is, they come with a skillset that is designed to manipulate the position and movement of other players (often the enemy, sometimes their own), and even sometimes outright incapacitating other players. They are tanks because they have low or 0 damage stun moves, displacement moves (i.e., players are forcibly moved against their will to an disadvantageous position), terrain creation/removal moves (blocks player movement), and typically high health and defensive stats.
Tanks in both cases can start fights because they have to pretty much. Their value is not in the damage, but in their ability to set up fights or to absorb damage/attention. For PVP in particular, tanks tend to have very heavy crowd control so you must be careful about if they decide to engage. The .5-2 seconds that is common for crowd control often is "just enough" to remove your main damage dealers from the fight (temporarily or permanently) and then you're at a disadvantage because the team numbers are against you.
For PVM, the tank wants to start its skill rotation so it can get specific buffs going and active.
ARPGs really don't have tanks btw (mostly RE:Diablo 2). You can certainly build them, but modern diablo forces all players to be damage dealing tanks. Diablo 3 tried to change this a little, but it still just ended up being an "Every stat is perfect" sort of game, and probably with more glass cannon builds.
The target of the enemies, and their tendency to attack, is usually modeled with "aggro". Damage increases aggro, but aggro can also be manipulated. Tanks usually have ways to increase their aggro, unproportionally to their damage, and DPSes sometimes have ways to lessen their aggro. Actually the tank's task is to take damage and manage aggro so that the enemy doesn't kill the DPS-es or the healers.
Because the roles of each fighter are baked into eg League of Legends, there are reasons tanks play the role of a tank. And yeah, opponents try to kill the other roles first, but the the job of a tank is to make that hard.