user21687: A friend and I tried playing this variant yesterday.  My question is:

Surely someone has looked at this before.  Does it have a standard name?  Or is there something obviously wrong with it that my friend and I missed?

Sometimes this kind of drastic change to the rules produces a game that is obviously no good.  (Consider the variant in which the board is turned into a torus; White has already lost, or perhaps wins immediately  with 1. QxK, depending on how you look at it.)  This variant didn't seem to have an obvious failure, and we had a good time playing one game and seeing how the tactics differed.
Specifically, the variant we played was:

All pieces move and capture the same as in standard chess, except:
Up to two pieces may occupy the same square.
A piece may move into an occupied square, but not through it.
A piece moving into a square occupied by two pieces of the opposite color may capture either, but not both.
Pieces of opposite colors sharing a square do not threaten one another.

So for example, it was legal for White to open with 1. Rh2, and have the rook and pawn sharing h2.  Having done this they could then continue on the following move with 2. Rh4.  But 1. Rh4 was
illegal.  
Since every chess variant imaginable has been tried out, I expect this one has also.  I'd like to read about it, but I don't know what to search for.

user20620: I did not find any information about this variation you are proposing which is interesting, but i do think it is equivalent to a 3D chess game where the board is 8x8x2 and the third dimension will be the "stacking" of the two pieces together on the same square.
Then when a piece moves, say for capturing one piece out of the two on the e5 square, it could choose between capturing on e5,1 or e5,2 .
By the way, just a thought on a thing to consider when playing it: if there is a situation where white has king on a1,1 and queen on b2,1 and black with a king on e5,1 and a knight on e5,2 - if Qxe5,2 then it will be a stalemate.

user21765: Pierre-Françoys Brousseau found a similar variant on Guy T Shafer's blog “Insane Chess”.   It says:                                                                                                                             

Crowded chess
Requires standard chess set & board. Pieces setup, and capture as normal.                                                                       
Pieces move as normal, except any 2 non-pawn pieces of the same
  color may share the same square. Men cannot pass through an
  occupied square: pieces are blocked by men as usual. But you may
  pass your own piece in 2 turns (one to move onto the occupied
  square, another to move off that square). Pawns may never share a
  space with any piece.

This is a little different from our variation, in which pieces of opposite colors may share space, and pawns may share space.
Unfortunately there is no real discussion there.

user18278: Ralph Betza has a series of "crowd" games, but they don't quite match what you have in mind.  The closest is the note in "Multiple Occupancy Miscellany", section "Crowd Capture":

What happens when you make a capturing move that ends on a square where there is more than one enemy piece? Many rules are possible.
The simple cases are that you capture

one enemy piece, chosen by the attacker (in this case multicolored crowds are common),
all enemy pieces, as in Troll Chess
All pieces, friend or foe,
No pieces at all.


https://www.chessvariants.com/boardrules.dir/multocc.html
https://www.chessvariants.com/boardrules.dir/crowded1.html
https://www.chessvariants.com/boardrules.dir/crowded2.html
https://www.chessvariants.com/boardrules.dir/fifo-chess.html 
Other Betza games where multi-occupancy shows up, but isn't perhaps the main attraction:
https://www.chessvariants.com/other.dir/pasgl312-chess.html
https://www.chessvariants.com/diffmove.dir/firefighter-chess.html

user21919: This sounds similar to a variant from the Netherlands known as Paco Ŝako ("peace chess"). In Paco Ŝako captured pieces are not removed from the board; instead, they become "merged" and occupy the same square. Merged pieces move as a unit according to the rules for the current player's piece, and cannot capture. However, they can be captured, in which case the capturing piece replaces the same-color piece from the merged pair, which then immediately moves to a new location. This allows a player to move multiple pieces in the same turn. Checkmate occurs when any piece merges with the opponent's king. This implies that the king cannot capture other pieces. Otherwise the rules for movement, castling, en passant, etc. work as in traditional chess. There is a YouTube video which illustrates the basic concepts.

21687: <p>A friend and I tried playing this variant yesterday.  My question is:</p>

<blockquote>
  <p>Surely someone has looked at this before.  Does it have a standard name?  Or is there something obviously wrong with it that my friend and I missed?</p>
</blockquote>

<p>Sometimes this kind of drastic change to the rules produces a game that is obviously no good.  (Consider the variant in which the board is turned into a torus; White has already lost, or perhaps wins immediately  with 1. QxK, depending on how you look at it.)  This variant didn't seem to have an obvious failure, and we had a good time playing one game and seeing how the tactics differed.</p>

<p>Specifically, the variant we played was:</p>

<ol>
<li>All pieces move and capture the same as in standard chess, except:</li>
<li>Up to two pieces may occupy the same square.</li>
<li>A piece may move <em>into</em> an occupied square, but not <em>through</em> it.</li>
<li>A piece moving into a square occupied by two pieces of the opposite color may capture either, but not both.</li>
<li>Pieces of opposite colors sharing a square do not threaten one another.</li>
</ol>

<p>So for example, it was legal for White to open with 1. Rh2, and have the rook and pawn sharing h2.  Having done this they could then continue on the following move with 2. Rh4.  But 1. Rh4 was
illegal.  </p>

<p>Since every chess variant imaginable has been tried out, I expect this one has also.  I'd like to read about it, but I don't know what to search for.</p>
 20620: <p>I did not find any information about this variation you are proposing which is interesting, but i do think it is equivalent to a 3D chess game where the board is <strong>8x8x2</strong> and the third dimension will be the "stacking" of the two pieces together on the same square.
Then when a piece moves, say for capturing one piece out of the two on the e5 square, it could choose between capturing on e5,1 or e5,2 .</p>

<p>By the way, just a thought on a thing to consider when playing it: if there is a situation where white has king on a1,1 and queen on b2,1 and black with a king on e5,1 and a knight on e5,2 - if Qxe5,2 then it will be a stalemate.</p>
 21765: <p>Pierre-Françoys Brousseau found <a href="https://boardgamegeek.com/blogpost/63635/crowded-chess" rel="nofollow noreferrer">a similar variant on Guy T Shafer's blog “Insane Chess”</a>.   It says:                                                                                                                             </p>

<blockquote>
  <h3>Crowded chess</h3>
  
  <p>Requires standard chess set &amp; board. Pieces setup, and capture as normal.                                                                       </p>
  
  <p>Pieces move as normal, except any 2 non-pawn pieces of the same
  color may share the same square. Men cannot pass through an
  occupied square: pieces are blocked by men as usual. But you may
  pass your own piece in 2 turns (one to move onto the occupied
  square, another to move off that square). Pawns may never share a
  space with any piece.</p>
</blockquote>

<p>This is a little different from our variation, in which pieces of opposite colors may share space, and pawns may share space.</p>

<p>Unfortunately there is no real discussion there.</p>
 18278: <p>Ralph Betza has a series of "crowd" games, but they don't quite match what you have in mind.  The closest is the note in "Multiple Occupancy Miscellany", section "Crowd Capture":</p>

<blockquote>
  <p>What happens when you make a capturing move that ends on a square where there is more than one enemy piece? Many rules are possible.</p>
  
  <p>The simple cases are that you capture</p>
  
  <ul>
  <li>one enemy piece, chosen by the attacker (in this case multicolored crowds are common),</li>
  <li>all enemy pieces, as in Troll Chess</li>
  <li>All pieces, friend or foe,</li>
  <li>No pieces at all.</li>
  </ul>
</blockquote>

<p><a href="https://www.chessvariants.com/boardrules.dir/multocc.html" rel="nofollow noreferrer">https://www.chessvariants.com/boardrules.dir/multocc.html</a><br>
<a href="https://www.chessvariants.com/boardrules.dir/crowded1.html" rel="nofollow noreferrer">https://www.chessvariants.com/boardrules.dir/crowded1.html</a><br>
<a href="https://www.chessvariants.com/boardrules.dir/crowded2.html" rel="nofollow noreferrer">https://www.chessvariants.com/boardrules.dir/crowded2.html</a><br>
<a href="https://www.chessvariants.com/boardrules.dir/fifo-chess.html" rel="nofollow noreferrer">https://www.chessvariants.com/boardrules.dir/fifo-chess.html</a>  </p>

<p>Other Betza games where multi-occupancy shows up, but isn't perhaps the main attraction:<br>
<a href="https://www.chessvariants.com/other.dir/pasgl312-chess.html" rel="nofollow noreferrer">https://www.chessvariants.com/other.dir/pasgl312-chess.html</a><br>
<a href="https://www.chessvariants.com/diffmove.dir/firefighter-chess.html" rel="nofollow noreferrer">https://www.chessvariants.com/diffmove.dir/firefighter-chess.html</a></p>
 21919: <p>This sounds similar to a variant from the Netherlands known as <a href="http://pacosako.com/" rel="nofollow noreferrer">Paco Ŝako</a> ("peace chess"). In Paco Ŝako captured pieces are not removed from the board; instead, they become "merged" and occupy the same square. Merged pieces move as a unit according to the rules for the current player's piece, and cannot capture. However, they can <em>be</em> captured, in which case the capturing piece replaces the same-color piece from the merged pair, which then immediately moves to a new location. This allows a player to move multiple pieces in the same turn. Checkmate occurs when any piece merges with the opponent's king. This implies that the king cannot capture other pieces. Otherwise the rules for movement, castling, <em>en passant</em>, etc. work as in traditional chess. There is a <a href="https://youtu.be/yJVcQK2gTdM" rel="nofollow noreferrer">YouTube video</a> which illustrates the basic concepts.</p>
