Setting datarrays with non-dimension coordinates errors

What happened?

I'm not sure if this is a bug or a feature but I was expecting this example to work since the new coord is just a slight rewrite of the original dimension coordinate:

import xarray as xr

ds = xr.tutorial.open_dataset("air_temperature")

# Change the first time value:
ds["air_new"] = ds.air.copy()
air_new_changed = ds.air_new[{"time": 0}] * 3
ds.air_new.loc[air_new_changed.coords] = air_new_changed  # Works! :)

# Add a another coord along time axis and change 
# the first time value:
ds["air_new"] = ds.air.copy().assign_coords(
    {"time_float": ds.time.astype(float)}
)
air_new_changed = ds.air_new[{"time": 0}] * 4
ds.air_new.loc[air_new_changed.coords] = air_new_changed  # Error! :(

Traceback (most recent call last):

  Cell In[25], line 5
    ds.air_new.loc[air_new_changed.coords] = air_new_changed

  File ~\AppData\Local\mambaforge\envs\jw\lib\site-packages\xarray\core\dataarray.py:222 in __setitem__
    dim_indexers = map_index_queries(self.data_array, key).dim_indexers

  File ~\AppData\Local\mambaforge\envs\jw\lib\site-packages\xarray\core\indexing.py:182 in map_index_queries
    grouped_indexers = group_indexers_by_index(obj, indexers, options)

  File ~\AppData\Local\mambaforge\envs\jw\lib\site-packages\xarray\core\indexing.py:144 in group_indexers_by_index
    raise KeyError(f"no index found for coordinate {key!r}")

KeyError: "no index found for coordinate 'time_float'"

Run the provided Python example with xarray to reproduce the KeyError. Start at DataArray.setitem and the indexing.py paths shown in the traceback, then verify that assignment using a non-dimension coordinate either works as expected or has its behavior clearly defined and tested.