BUG: Can only compare identically-labeled Series objects (string vs. object)

[wahsmail]

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Reproducible Example

s1 = pd.Series([1, 2, 3], index=['a', 'b', 'c'])
s2 = pd.Series([4, 5, 6], index=['a', 'b', 'c'])
s2.index = s2.index.astype('string')

s1 < s2  # fails

s1, s2 = s1.align(s2)
s1 < s2  # also fails

s1 = s1.reindex(s2.index)
s1 < s2  # succeeds

Issue Description

When a series (or dataframe) with otherwise identical indices are compared, but the indexes are technically dtype(object) and dtype(string), element-wise comparison fails. In the debugger, it looks like the ExtensionArray StringArray.equals is False when comparing to a python list of strings, causing Series._indexed_same to return False.

Expected Behavior

Ideally the string and object dtype would be comparable. This in-between state for Pandas dtypes has been quite awkward, with some libraries porting over to numpy-nullable / pyarrow dtype backends, but the Pandas library defaults not using them yet.

Installed Versions

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