ENH: Inference for ExtensionArray and ExtensionDtype.type

[Xyhlon]

Is your feature request related to a problem?

I wish I could use pandas to do the conversion of ExtensionDtype and ExtensionArray automatically like it does with the internal Pandas Dtypes like eg. Period Intervall with their respective Dtype.

    test_uarray = [ufloat(10, 1), ufloat(3.8, 0.2)]
    test_periods = [Period("4Q2005")] * len(test_uarray)
    df_uarray = pd.DataFrame(
        {
            "ufloats": test_uarray,
            "periods": test_periods,
        }
    )
    print(df_uarray.dtypes)

output should be

ufloats           ufloat
ints               int64
periods    period[Q-DEC]
dtype: object

Where Ufloat is the registered corresponding ExtensionDtype and having a corresponding ExtensionArray UfloatArray.
[this should provide a description of what the problem is, e.g. "I wish I could use pandas to do [...]"]

Describe the solution you'd like

For this to work infer_dtype_from_scalar() needs to be patched.

A possible patch would be adding the following here

    from pandas.core.dtypes.base import _registry

    for extdtype in _registry.dtypes:
        if isinstance(extdtype.type, property):
            try:
                exttype = extdtype.type.fget(extdtype.type)
            except AttributeError:
                continue
        else:
            exttype = extdtype.type

        if isinstance(val, exttype):
            dtype = extdtype()

With this patch inference of registered ExtensionDtype is possible

Furthermore the Inference of ExtensionArray needs to be fixed.
This can be accomplished by patching sanitize_array

The following lines can be added to sanitize_array here in order to add the functionality, however I wouldn't recommend doing it that way.

                inferred, _ = infer_dtype_from_scalar(val=subarr[0], pandas_dtype=True)
                subarr = maybe_cast_to_extension_array(
                    inferred.construct_array_type(), subarr, inferred
                )

Furthermore these two functions need to be imported
infer_dtype_from_scalar,
maybe_cast_to_extension_array,

API breaking implications

This should affect the API it only uses the already requirements to the ExtensionDtype and ExtensionArray

[mocquin]

Duplicate of #41848 ?

[mocquin]

As well as #27995 ?