@chesser4444: Better post your question in chess variants forum. btw there are some variants like 'synchronous chess' and 'parity chess' that seem to work that way. just google for these words. you will find materials on them.
first move advantage

Having the players move their pieces simultaneously, however that might work (do we write our moves down on a piece of paper hidden from view from our opponent while they do the same?), would radically change the game to be unrecognizable as chess. Another idea I had on this topic (not claiming I'm the only one who has thought of this) was that white makes the first move, then on subsequent turns each player makes two moves. I don't know which side that would favor. After each turn, the side that just moved has made one more move than their opponent. But this is also a radical departure from chess.
A more practical idea that preserves the essential gameplay of chess would be to restrict white's first move options. How much of an advantage would white have, if any, when we disallow the four most common opening moves (e4, d4, c4, Nf3)? Just on the first move. White can make any move as normal on subsequent moves. My guess is that would result in fairly even win percentages.

what if they both want to land on one square
Winner of the best of three 'Paper/Scissors/Rock' bouts gets the square. Obviously.

But seriously, white's first move option (and advantage) can be restricted without any outside or advance judgement of specific moves, such as the 4 most common that JSLigon mentions.
The way to do that is have player #1, who is NOT guaranteed to keep the white side, make white's first move. Only after seeing that move does player #2 choose whether to take black or white.
Player # 1 will not make white's first move one that they think is the best, because then they will end up playing black against it, so no 1. d4 or 1. Nf3, etc.
One way to do this would be to make the players switch seats if player #2 so choses, or rotate the board. Another way would be to let black retract white's first move, ONLY IF they then go with the symmetrical reply. For example, if white playes 1. e2-e4, black could move the white pawn from e4 back to e2 and then play e7-e5, but no other move.
If player #1 choses something that is actually bad, such as 1. f2-f3, player #2 can just keep black and continue as normal...

I like that idea a lot. Best strategy for the first player would be to make the opening move that they think leads to the most even game. 1 e4 or 1 d4 wouldn't normally be chosen, but neither would 1 g4 or 1 h4 since then the first player would get stuck playing a presumably disadvantageous position with white. But if someone happens to like playing black against 1 e4, they could open with that too. No artificial move restrictions.
I read some articles about first move advantage to white. Some statistics seem to show that white has a small advantage in starting first. I am wondering if the players move their pieces simultaneously then this advantage can be eliminated. Is there any such variant of chess ?
thx.