I have searched the Web for various Pokémon multiplayer variants. I am documenting my findings in this page.
Free-for-all
I have come across a set of rules to play a multiplayer version of the game (see 3-Player | BoardGameGeek). In this free-for-all version in which the winner is the first player to draw all of their prizes, players can only attack to the right. However, they can decide to allocate a portion of the damage to the next player counter-clockwise by reducing by half the excess damage (rounded up). So for instance, if an attack did 200 damage, the attacking player could decide to allocate 100 damage to the right and the remaining 100 damage could be applied to the active Pokémon of the player to the right of the right player. The remaining damage would be halved to 50 damage in this case. If there were a fourth player, the excess damage could be distributed to all three other players, the remaining damage being halved each time. In this format, special effects (like not being able to use item cards until the next turn due to Dragonite's "Deafen" attack) can only be applied on player to the right. Another option is to only allow attacking to the left. This would more resemble 1v1 play, but not be as engaging.
Companion
This format is similar to the MTG Commander format. Each player creates a hundred-card singleton deck that contains a Companion (one Basic Pokémon and up to one Stage 1 and up to one Stage 2 Pokémon from this evolution line, each Pokémon complying with some predetermined HP limit. All cards in the deck must share the color identity of the Companion (energy symbol on the Companion card(s) other than Weakness and Resistance). If the Companion only has Fighting symbols on their cards, then all other cards in the deck must only contain Fighting energy symbols (including Trainer cards).
At the start of the game, each player puts the Companion in the Companion Zone and draws seven cards. Mulligans are resolved. Each player then puts 10 prizes in front of them from their library. The winner is the player that first draws all of their prizes.
The full Companion rule guide can be found here.
Team Format
The "Pokémon TCG Alternative Play Handbook" that can be found in the Rules & Resouces page accessed from the Pokémon Trading Card Game Rulebook | Pokemon.com page describes a 2v2 format called "Team Format". Here is a summary of the rules for this format.
Set Up
Both players of the same team sit on the same side of the table. Each team's space is divided in three lanes. One for each player and a common lane shared by both players located between them. Each player brings a 60-card deck to the game, sleeved in a way that makes it easy to recognize the card's owner between the two players of the same team. After drawing seven cards, placing the active Pokémon in their lane, adding up to five Pokémon to their bench and resolving mulligans, each player draws six cards from their library, which become the player's six prizes for the game. Each player lays out three prizes on both side of their lane. Each lane will thus have 6 prizes (the center lane's prizes are shared - three from each player). No Pokémon is played in the center lane at set up time.
Gameplay
As a team, both players are taking their turns together, resolving Special Conditions at the same time, attacking together and passing together. Like a normal game, a player can only play one Supporter per turn and one energy per turn in each lane.
Any cards played can only affect the lane they are played in.
There is no deck in the center lane. Both player can play cards in the center lane. Energies can be attached to this lane by either player, but only once per turn. Only one Supporter card can be played in the center lane, by either player. Cards played in the center lane that target an opponent can target either opponent, but nor both. When a card played in the center lane needs to enter a player's hand, Discard pile, Lost Zone or any other zone, it enters its owner's corresponding zone. Abilities of Pokémon in the center lane can be used by players only one time, unless the card says otherwise. When an ability can only be used one time, either player can use it, but not both. When an ability can be used multiple times, it can be used by both players.
Pokémon can be played in the player's lane or in the center lane. The active spot in a lane must be filled before any Pokémon can be played on the bench.
Each lane can have a Stadium card. Effects of Stadium cards only affect the lane they are in.
When a card references prizes, it refers to prizes to the left and to the right of the lane where the card is played.
Adjustments to Attacking
If an opponent swings into a lane that does not have any active or benched Pokémon, then as soon as the attack does at least 10 damage, the opponent can claim a prize.
If a Pokémon is knocked out in the center lane, the team decides which player takes the prize. If more than one prize needs to be taken, then the team decides how to allocate prizes.
Prizes can only be taken in the lane where the action leading to taking the prize took place. If there are no prizes left, no prizes are taken. Attacks can still take place in this lane and are resolved as normal, except no prizes are awarded.
My rule interpretation is that the first player cannot attack first turn in the center lane.
Loss Conditions
If a player requiring to draw a card is unable to do so, they are eliminated from the game. They remove all cards in their player's lane, but leave any cards they own in the center lane. The opposing team can then attack their now empty lane and take prizes as soon as they perform at least 10 damage (placing damage counter does not count).
If a team has no Pokémon in play in any lane, they lose the game.
Winning the Game
The first team to redeem 9 prizes is the winner.
I found a video on YouTube that describes this 2v2 format (see How to Play Pokemon TCG 2v2 Format).