id,node_id,number,title,user,state,locked,assignee,milestone,comments,created_at,updated_at,closed_at,author_association,active_lock_reason,draft,pull_request,body,reactions,performed_via_github_app,state_reason,repo,type
783272657,MDU6SXNzdWU3ODMyNzI2NTc=,4791,Assigning non-dimension coordinates does not conserve MultiIndex type,34008825,open,0,,,3,2021-01-11T11:03:00Z,2023-08-23T13:18:15Z,,NONE,,,,"<!-- Please include a self-contained copy-pastable example that generates the issue if possible.

Please be concise with code posted. See guidelines below on how to provide a good bug report:

- Craft Minimal Bug Reports: http://matthewrocklin.com/blog/work/2018/02/28/minimal-bug-reports
- Minimal Complete Verifiable Examples: https://stackoverflow.com/help/mcve

Bug reports that follow these guidelines are easier to diagnose, and so are often handled much more quickly.
-->

**What happened**:
Assigning a new non-dimension coordinate of type `MultiIndex` to a `Dataset/DataArray` returns an object that has the new coordinate as type `Object` instead of `MultiIndex`.

**What you expected to happen**:
Adding a `MultiIndex` as a dimension coordinate to a `Dataset` keeps the type. It should be the same for non-dimension.

**Minimal Complete Verifiable Example**:

```python
import numpy as np
import pandas as pd
import xarray as xr

ds = xr.Dataset(
    data_vars={'foo': (['x', 'y'], np.random.rand(2, 2))},
    coords={'x': [1, 2], 'y': [1, 2]}
)
idx = pd.MultiIndex.from_arrays([['a', 'b'], [5, 6]])

ds.assign_coords({'z': idx}) # Conserves MultiIndex type
ds.assign_coords({'z': ('x', idx)}) # Does not conserve MultiIndex type
```
**Anything else we need to know?**:

**Environment**:

<details><summary>Output of <tt>xr.show_versions()</tt></summary>

INSTALLED VERSIONS
------------------
commit: None
python: 3.7.9 (default, Aug 31 2020, 17:10:11) [MSC v.1916 64 bit (AMD64)]
python-bits: 64
OS: Windows
OS-release: 10
machine: AMD64
processor: Intel64 Family 6 Model 142 Stepping 12, GenuineIntel
byteorder: little
LC_ALL: None
LANG: en
LOCALE: None.None
libhdf5: 1.10.4
libnetcdf: 4.7.3

xarray: 0.16.1
pandas: 1.1.3
numpy: 1.19.1
scipy: 1.5.2
netCDF4: 1.5.3
pydap: installed
h5netcdf: None
h5py: None
Nio: None
zarr: 2.6.1
cftime: 1.3.0
nc_time_axis: None
PseudoNetCDF: None
rasterio: None
cfgrib: None
iris: None
bottleneck: None
dask: 2.30.0
distributed: 2.30.1
matplotlib: 3.3.3
cartopy: 0.18.0
seaborn: 0.11.0
numbagg: None
pint: None
setuptools: 50.3.0.post20201006
pip: 20.2.4
conda: 4.9.2
pytest: None
IPython: 7.18.1
sphinx: 3.2.1


</details>
","{""url"": ""https://api.github.com/repos/pydata/xarray/issues/4791/reactions"", ""total_count"": 0, ""+1"": 0, ""-1"": 0, ""laugh"": 0, ""hooray"": 0, ""confused"": 0, ""heart"": 0, ""rocket"": 0, ""eyes"": 0}",,,13221727,issue
778863478,MDU6SXNzdWU3Nzg4NjM0Nzg=,4765,Conversions to Pandas objects fail when mixtures of Index and MultiIndex are present,34008825,closed,0,,,3,2021-01-05T10:49:25Z,2021-06-26T18:24:46Z,2021-06-26T18:24:46Z,NONE,,,,"Hello Xarray team. Thank you very much for your efforts, they have made my job more pleasant.
<!-- Please include a self-contained copy-pastable example that generates the issue if possible.

Please be concise with code posted. See guidelines below on how to provide a good bug report:

- Craft Minimal Bug Reports: http://matthewrocklin.com/blog/work/2018/02/28/minimal-bug-reports
- Minimal Complete Verifiable Examples: https://stackoverflow.com/help/mcve

Bug reports that follow these guidelines are easier to diagnose, and so are often handled much more quickly.
-->

**What happened**:
When I try to convert a DataArray with a mix of MultiIndex and Index into any pandas object, it returns `NotImplementedError: isna is not defined for MultiIndex`.

**What you expected to happen**:
If the DataArray contains all simple Index objects or one single MultiIndex, the transformation succeeds. I would expect the same from a mixture.

**Minimal Complete Verifiable Example**:

```python
import numpy as np
import pandas as pd
import xarray as xr

da = xr.DataArray(
    np.random.randint(0, 10, size=[3, 3, 3]),
    coords={'x': ['x1', 'x2', 'x3'], 'y': ['y1', 'y2', 'y3'], 'z': ['z1', 'z2', 'z3']},
    dims=['x', 'y', 'z']
)

da.to_series() # This works

da.stack(xyz=['x', 'y', 'z']).to_series() # This works

da.stack(xy=['x', 'y']).to_series() # This does not work
```

**Anything else we need to know?**:
It happens the same thing with DataSet.

**Environment**:

<details><summary>Output of <tt>xr.show_versions()</tt></summary>
INSTALLED VERSIONS
------------------
commit: None
python: 3.7.9 (default, Aug 31 2020, 17:10:11) [MSC v.1916 64 bit (AMD64)]
python-bits: 64
OS: Windows
OS-release: 10
machine: AMD64
processor: Intel64 Family 6 Model 142 Stepping 12, GenuineIntel
byteorder: little
LC_ALL: None
LANG: en
LOCALE: None.None
libhdf5: 1.10.4
libnetcdf: 4.7.3

xarray: 0.16.1
pandas: 1.1.3
numpy: 1.19.1
scipy: 1.5.2
netCDF4: 1.5.3
pydap: installed
h5netcdf: None
h5py: None
Nio: None
zarr: None
cftime: 1.3.0
nc_time_axis: None
PseudoNetCDF: None
rasterio: None
cfgrib: None
iris: None
bottleneck: None
dask: 2.30.0
distributed: 2.30.1
matplotlib: 3.3.3
cartopy: 0.18.0
seaborn: 0.11.0
numbagg: None
pint: None
setuptools: 50.3.0.post20201006
pip: 20.2.4
conda: 4.9.2
pytest: None
IPython: 7.18.1
sphinx: 3.2.1

</details>
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