BUG: `assert_index_equal(CategoricalIndex, CategoricalIndex, check_categorical=True, exact=False)` raises TypeError instead of AssertionError

[mroeschke]

Pandas version checks

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• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

In [1]: import pandas as pd

In [2]: p_left = pd.Index([1, 2, 3], name="a", dtype="category")

In [3]: p_right = pd.Index([1, 2, 6], name="a", dtype="category")

In [4]: pd.testing.assert_index_equal(p_left, p_right, check_categorical=True, exact=False)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[10], line 1
----> 1 pd.testing.assert_index_equal(p_left, p_right, check_categorical=True, exact=False)

    [... skipping hidden 1 frame]

File ~/pandas/core/ops/common.py:70, in _unpack_zerodim_and_defer.<locals>.new_method(self, other)
     66         return NotImplemented
     68 other = item_from_zerodim(other)
---> 70 return method(self, other)

File ~/pandas/core/arrays/categorical.py:143, in _cat_compare_op.<locals>.func(self, other)
    141 msg = "Categoricals can only be compared if 'categories' are the same."
    142 if not self._categories_match_up_to_permutation(other):
--> 143     raise TypeError(msg)
    145 if not self.ordered and not self.categories.equals(other.categories):
    146     # both unordered and different order
    147     other_codes = recode_for_categories(
    148         other.codes, other.categories, self.categories, copy=False
    149     )

TypeError: Categoricals can only be compared if 'categories' are the same.

Issue Description

pandas asserters should always raise an AssertionError.

Expected Behavior

I would expect this to raise an AssertionError.

Installed Versions

Replace this line with the output of pd.show_versions()