add editable html row with javascript and HTML only

As others noted the cells variables are <td> elements already. You can just set the contentEditable property to true.

Also I would change the mouseup event to a click event. The mouseup event is not triggered in keyboard inputs. However if you want to use the mouseup for some reason you can still do it.

document.getElementById("addRow").addEventListener("click", function() {

  var table = document.getElementById("table")
  var row = table.insertRow(0);

  var cell0 = row.insertCell(0);
  cell0.contentEditable = true;
  cell0.textContent = 'Property';

  var cell1 = row.insertCell(1);
  cell1.contentEditable = true;
  cell1.textContent = 'Value'
})

table,
th,
td {
  border: 1px solid black;
}

<div id="wrapper">
  <button id="addRow">+</button>
  <button id="deleteRow">-</button>
  <table id="table">
    <tbody>
      <!-- filled by script -->
    </tbody>
  </table>
</div>

The problem isn't contenteditable, the problem is that the code is trying to append a <td> element immediately within a <td> element. cell0 and cell1 already are <td> elements. The browser is "correcting" the markup and removing the invalid inner <td>, leaving you with just a table with text in it.

Change those inner <td> elements to something like a <div> instead:

cell0.innerHTML = '<div contenteditable="true">Property</div>';
cell1.innerHTML = '<div contenteditable="true">Value</div>'