The solution that makes sense to me is:

Multiindex level name conflicts should only be checked for coordinates, not data variables.

But I've only spent a few hours digging through the codebase to try and understand this problem - I'm not quite sure what the implications would be.

Here is another place where it feels like it makes more sense to only check the MultiIndex level names of coords:

```python

da = xr.DataArray([0, 1], dims=["location"], coords={"lat": ("location", [10, 11]), "lon": ("location", [20, 21])}).set_index(location=["lat", "lon"]) location = da["location"]

you cannot directly make a dataset with location as a data variable

xr.Dataset({"data": location}) Traceback (most recent call last): File "<stdin>", line 1, in <module> 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' (data) 'lon' (location), 'lon' (data)

but if you go a round-about way, you can exploit that assign_coords only checks

the multiindex names of coordinates, not data variables

```python

ds = xr.Dataset({"data": xr.DataArray(data=location.variable._data, dims=["location"])}) ds = ds.assign_coords({"location": location}) ds <xarray.Dataset> Dimensions: (location: 2) Coordinates: * location (location) MultiIndex - lat (location) int64 10 11 - lon (location) int64 20 21 Data variables: data (location) object (10, 20) (11, 21) ds["data"].variable._data PandasIndexAdapter(array=MultiIndex([(10, 20), (11, 21)], names=['lat', 'lon']), dtype=dtype('O')) ```

If making variable.assert_unique_multiindex_level_names only check coords is the way to go, I'm keen + happy to try putting together a pull request for this.