Error concatenating Multiindex variables

MCVE Code Sample

>>> import xarray as xr
>>> da = xr.DataArray([0, 1], dims=["location"], coords={"lat": ("location", [10, 11]), "lon": ("location", [20, 21])}).set_index(location=["lat", "lon"])
>>> da2 = xr.DataArray([2, 3], dims=["location"], coords={"lat": ("location", [12, 13]), "lon": ("location", [22, 23])}).set_index(location=["lat", "lon"])
>>> xr.concat([da["location"], da2["location"]], dim="location")
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/harry/code/xarray/xarray/core/concat.py", line 135, in concat
    return f(objs, dim, data_vars, coords, compat, positions, fill_value, join)
  File "/home/harry/code/xarray/xarray/core/concat.py", line 431, in _dataarray_concat
    ds = _dataset_concat(
  File "/home/harry/code/xarray/xarray/core/concat.py", line 384, in _dataset_concat
    result = Dataset(result_vars, attrs=result_attrs)
  File "/home/harry/code/xarray/xarray/core/dataset.py", line 541, in __init__
    variables, coord_names, dims, indexes = merge_data_and_coords(
  File "/home/harry/code/xarray/xarray/core/merge.py", line 466, in merge_data_and_coords
    return merge_core(
  File "/home/harry/code/xarray/xarray/core/merge.py", line 556, in merge_core
    assert_unique_multiindex_level_names(variables)
  File "/home/harry/code/xarray/xarray/core/variable.py", line 2363, in assert_unique_multiindex_level_names
    raise ValueError("conflicting MultiIndex level name(s):\n%s" % conflict_str)
ValueError: conflicting MultiIndex level name(s):
'lat' (location), 'lat' (<this-array>)
'lon' (location), 'lon' (<this-array>)

Expected Output

The output should be the same as first concatenating the DataArrays, then extracting the dimension location:

>>> xr.concat([da, da2], dim="location")["location"]
<xarray.DataArray 'location' (location: 4)>
array([(10, 20), (11, 21), (12, 22), (13, 23)], dtype=object)
Coordinates:
  * location  (location) MultiIndex
  - lat       (location) int64 10 11 12 13
  - lon       (location) int64 20 21 22 23

Problem Description

>>> # da["location"] looks like a normal DataArray
>>> location = da["location"]
>>> location
<xarray.DataArray 'location' (location: 2)>
array([(10, 20), (11, 21)], dtype=object)
Coordinates:
  * location  (location) MultiIndex
  - lat       (location) int64 10 11
  - lon       (location) int64 20 21
>>> # but in actual fact, the variable._data is a MultiIndex
>>> location.variable._data
PandasIndexAdapter(array=MultiIndex([(10, 20),
            (11, 21)],
           names=['lat', 'lon']), dtype=dtype('O'))

This is why an error is thrown: variable.assert_unique_multiindex_level_names gets passed two variables: location.variable (the DataArray data values), and also location["location"].variable (the coordinate values), which are both MultiIndexes.

Output of xr.show_versions()

Reproduce the MCVE with the concatenation path in xarray/core/concat.py, then inspect the merge and MultiIndex checks in xarray/core/merge.py and xarray/core/variable.py. Add or update a regression test for concatenating the extracted location DataArrays; done means it produces the four expected MultiIndex values without the conflicting-level-name error.