xr.testing.assert_equal does not test for dtype

In #4622 @toddrjen points out that xr.testing.assert_equal does not test for the dtype, only for the value. Therefore the following does not raise an error:

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

xr.testing.assert_equal(
    xr.DataArray(np.array(1, dtype=int)), xr.DataArray(np.array(1, dtype=float))
)
xr.testing.assert_equal(
    xr.DataArray(np.array(1, dtype=int)), xr.DataArray(np.array(1, dtype=object))
)
xr.testing.assert_equal(
    xr.DataArray(np.array("a", dtype=str)), xr.DataArray(np.array("a", dtype=object))
)

This comes back to numpy, i.e. the following is True:

np.array(1, dtype=int) == np.array(1, dtype=float)

Depending on the situation one or the other is desirable or not. Thus, I would suggest to add a check_dtype argument to xr.testing.assert_equal and also to DataArray.equals (and Dataset and Variable and identical). I have not seen such an option in numpy, but pandas has it (e.g. pd.testing.assert_series_equal(left, right, check_dtype=True, ...) . I would not change __eq__.

• Thoughts?
• What should the default be? We could try True first and see how many failures this creates?
• What to do with coords and indexes? pd.testing.assert_series_equal has a check_index_type keyword. Probably we need check_coords_type as well? This makes the whole thing much more complicated... Also #4543

Start by reviewing xr.testing.assert_equal and the equals and identical methods on DataArray, Dataset, and Variable, then reproduce the dtype examples from the issue. Determine the intended check_dtype default and how coordinate and index types should be handled; the work is done when the API behavior and those choices are consistently defined and verified.