Handling of non-string dimension names

While working on a pull request (#5149) for #5146 I came across an inconsistency in allowed dimension names. If I try and create a DataArray with a non-string dimension, I get a TypeError:

>>> import xarray as xr
>>> da = xr.DataArray(np.ones((5, 5)), dims=[1, "y"])
...
TypeError: dimension 1 is not a string

But creating it with a string and renaming it works:

>>> da = xr.DataArray(np.ones((5, 5)), dims=["x", "y"]).rename(x=1)
>>> da
<xarray.DataArray (1: 5, y: 5)>
array([[1., 1., 1., 1., 1.],
       [1., 1., 1., 1., 1.],
       [1., 1., 1., 1., 1.],
       [1., 1., 1., 1., 1.],
       [1., 1., 1., 1., 1.]])
Dimensions without coordinates: 1, y

I can create a dataset via this renaming, but trying to get the repr value fails as xarray.core.utils.SortedKeysDict tries to sort it and cannot compare the string dimension to the int dimension:

>>> import xarray as xr
>>> ds = xr.Dataset({"test": xr.DataArray(np.ones((5, 5)), dims=["x", "y"]).rename(x=1)})
>>> ds
...
~/software/external/xarray/xarray/core/formatting.py in dataset_repr(ds)
    519 
    520     dims_start = pretty_print("Dimensions:", col_width)
--> 521     summary.append("{}({})".format(dims_start, dim_summary(ds)))
    522 
    523     if ds.coords:

~/software/external/xarray/xarray/core/formatting.py in dim_summary(obj)
    422 
    423 def dim_summary(obj):
--> 424     elements = [f"{k}: {v}" for k, v in obj.sizes.items()]
    425     return ", ".join(elements)
    426 

~/software/external/xarray/xarray/core/formatting.py in <listcomp>(.0)
    422 
    423 def dim_summary(obj):
--> 424     elements = [f"{k}: {v}" for k, v in obj.sizes.items()]
    425     return ", ".join(elements)
    426 

/usr/lib/python3.9/_collections_abc.py in __iter__(self)
    847 
    848     def __iter__(self):
--> 849         for key in self._mapping:
    850             yield (key, self._mapping[key])
    851 

~/software/external/xarray/xarray/core/utils.py in __iter__(self)
    437 
    438     def __iter__(self) -> Iterator[K]:
--> 439         return iter(self.mapping)
    440 
    441     def __len__(self) -> int:

~/software/external/xarray/xarray/core/utils.py in __iter__(self)
    504     def __iter__(self) -> Iterator[K]:
    505         # see #4571 for the reason of the type ignore
--> 506         return iter(sorted(self.mapping))  # type: ignore[type-var]
    507 
    508     def __len__(self) -> int:

TypeError: '<' not supported between instances of 'str' and 'int'

The same thing happens if I call rename on the dataset rather than the array it is initialised with.

If the initialiser requires the dimension names to be strings, and other code (which includes the HTML formatter I was looking at when I found this) assume that they are, then rename and any other method which can alter dimension names should also enforce the string requirement.

Environment:

Start with dimension validation in the DataArray initializer and the rename paths. Inspect formatting.py's dim_summary and utils.py's SortedKeysDict, which currently fail when string and integer dimension names are mixed. Done means dimension names are handled consistently: invalid non-string names are rejected by rename paths, and dataset repr no longer fails for this case.