diff --git a/pandas/_libs/sparse.pyx b/pandas/_libs/sparse.pyx
index 321d7c374d8ec..0c3d8915b749b 100644
--- a/pandas/_libs/sparse.pyx
+++ b/pandas/_libs/sparse.pyx
@@ -103,7 +103,7 @@ cdef class IntIndex(SparseIndex):
         if not monotonic:
             raise ValueError("Indices must be strictly increasing")
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         if not isinstance(other, IntIndex):
             return False
 
@@ -399,7 +399,7 @@ cdef class BlockIndex(SparseIndex):
             if blengths[i] == 0:
                 raise ValueError(f'Zero-length block {i}')
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         if not isinstance(other, BlockIndex):
             return False
 
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index 921927325a144..d85647edc3b81 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -7,7 +7,7 @@
    without warning.
 """
 import operator
-from typing import Any, Callable, Dict, Optional, Sequence, Tuple, Union
+from typing import Any, Callable, Dict, Optional, Sequence, Tuple, Union, cast
 
 import numpy as np
 
@@ -20,7 +20,12 @@
 from pandas.util._validators import validate_fillna_kwargs
 
 from pandas.core.dtypes.cast import maybe_cast_to_extension_array
-from pandas.core.dtypes.common import is_array_like, is_list_like, pandas_dtype
+from pandas.core.dtypes.common import (
+    is_array_like,
+    is_dtype_equal,
+    is_list_like,
+    pandas_dtype,
+)
 from pandas.core.dtypes.dtypes import ExtensionDtype
 from pandas.core.dtypes.generic import ABCDataFrame, ABCIndexClass, ABCSeries
 from pandas.core.dtypes.missing import isna
@@ -742,7 +747,7 @@ def searchsorted(self, value, side="left", sorter=None):
         arr = self.astype(object)
         return arr.searchsorted(value, side=side, sorter=sorter)
 
-    def equals(self, other: "ExtensionArray") -> bool:
+    def equals(self, other: object) -> bool:
         """
         Return if another array is equivalent to this array.
 
@@ -762,7 +767,8 @@ def equals(self, other: "ExtensionArray") -> bool:
         """
         if not type(self) == type(other):
             return False
-        elif not self.dtype == other.dtype:
+        other = cast(ExtensionArray, other)
+        if not is_dtype_equal(self.dtype, other.dtype):
             return False
         elif not len(self) == len(other):
             return False
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index 6e5c7bc699962..a28b341669918 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -2242,7 +2242,7 @@ def _from_factorized(cls, uniques, original):
             original.categories.take(uniques), dtype=original.dtype
         )
 
-    def equals(self, other):
+    def equals(self, other: object) -> bool:
         """
         Returns True if categorical arrays are equal.
 
@@ -2254,7 +2254,9 @@ def equals(self, other):
         -------
         bool
         """
-        if self.is_dtype_equal(other):
+        if not isinstance(other, Categorical):
+            return False
+        elif self.is_dtype_equal(other):
             if self.categories.equals(other.categories):
                 # fastpath to avoid re-coding
                 other_codes = other._codes
diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index fcb7e2a949205..651c079ecc08e 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -22,6 +22,7 @@
     Tuple,
     Type,
     Union,
+    cast,
 )
 import warnings
 import weakref
@@ -1195,7 +1196,7 @@ def _indexed_same(self, other) -> bool:
             self._get_axis(a).equals(other._get_axis(a)) for a in self._AXIS_ORDERS
         )
 
-    def equals(self, other):
+    def equals(self, other: object) -> bool:
         """
         Test whether two objects contain the same elements.
 
@@ -1275,6 +1276,7 @@ def equals(self, other):
         """
         if not (isinstance(other, type(self)) or isinstance(self, type(other))):
             return False
+        other = cast(NDFrame, other)
         return self._mgr.equals(other._mgr)
 
     # -------------------------------------------------------------------------
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index ecd3670e724a1..623ce68201492 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -4176,7 +4176,7 @@ def putmask(self, mask, value):
             # coerces to object
             return self.astype(object).putmask(mask, value)
 
-    def equals(self, other: Any) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determine if two Index object are equal.
 
diff --git a/pandas/core/indexes/category.py b/pandas/core/indexes/category.py
index fb283cbe02954..4990e6a8e20e9 100644
--- a/pandas/core/indexes/category.py
+++ b/pandas/core/indexes/category.py
@@ -290,7 +290,7 @@ def _is_dtype_compat(self, other) -> bool:
 
         return other
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determine if two CategoricalIndex objects contain the same elements.
 
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 8ccdab21339df..6d9d75a69e91d 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -24,7 +24,7 @@
     is_scalar,
 )
 from pandas.core.dtypes.concat import concat_compat
-from pandas.core.dtypes.generic import ABCIndex, ABCIndexClass, ABCSeries
+from pandas.core.dtypes.generic import ABCIndex, ABCSeries
 
 from pandas.core import algorithms
 from pandas.core.arrays import DatetimeArray, PeriodArray, TimedeltaArray
@@ -130,14 +130,14 @@ def __array_wrap__(self, result, context=None):
 
     # ------------------------------------------------------------------------
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determines if two Index objects contain the same elements.
         """
         if self.is_(other):
             return True
 
-        if not isinstance(other, ABCIndexClass):
+        if not isinstance(other, Index):
             return False
         elif not isinstance(other, type(self)):
             try:
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index 9548ebbd9c3b2..e8d0a44324cc5 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -1005,19 +1005,20 @@ def _format_space(self) -> str:
     def argsort(self, *args, **kwargs) -> np.ndarray:
         return np.lexsort((self.right, self.left))
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determines if two IntervalIndex objects contain the same elements.
         """
         if self.is_(other):
             return True
 
-        # if we can coerce to an II
-        # then we can compare
+        # if we can coerce to an IntervalIndex then we can compare
         if not isinstance(other, IntervalIndex):
             if not is_interval_dtype(other):
                 return False
             other = Index(other)
+            if not isinstance(other, IntervalIndex):
+                return False
 
         return (
             self.left.equals(other.left)
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 13927dede5542..9ca3075d2962c 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -3227,7 +3227,7 @@ def truncate(self, before=None, after=None):
             verify_integrity=False,
         )
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determines if two MultiIndex objects have the same labeling information
         (the levels themselves do not necessarily have to be the same)
@@ -3270,11 +3270,10 @@ def equals(self, other) -> bool:
                 np.asarray(other.levels[i]._values), other_codes, allow_fill=False
             )
 
-            # since we use NaT both datetime64 and timedelta64
-            # we can have a situation where a level is typed say
-            # timedelta64 in self (IOW it has other values than NaT)
-            # but types datetime64 in other (where its all NaT)
-            # but these are equivalent
+            # since we use NaT both datetime64 and timedelta64 we can have a
+            # situation where a level is typed say timedelta64 in self (IOW it
+            # has other values than NaT) but types datetime64 in other (where
+            # its all NaT) but these are equivalent
             if len(self_values) == 0 and len(other_values) == 0:
                 continue
 
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 3577a7aacc008..6080c32052266 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -433,7 +433,7 @@ def argsort(self, *args, **kwargs) -> np.ndarray:
         else:
             return np.arange(len(self) - 1, -1, -1)
 
-    def equals(self, other) -> bool:
+    def equals(self, other: object) -> bool:
         """
         Determines if two Index objects contain the same elements.
         """
diff --git a/pandas/core/internals/managers.py b/pandas/core/internals/managers.py
index aa74d173d69b3..371b721f08b27 100644
--- a/pandas/core/internals/managers.py
+++ b/pandas/core/internals/managers.py
@@ -1437,7 +1437,10 @@ def take(self, indexer, axis: int = 1, verify: bool = True, convert: bool = True
             new_axis=new_labels, indexer=indexer, axis=axis, allow_dups=True
         )
 
-    def equals(self, other: "BlockManager") -> bool:
+    def equals(self, other: object) -> bool:
+        if not isinstance(other, BlockManager):
+            return False
+
         self_axes, other_axes = self.axes, other.axes
         if len(self_axes) != len(other_axes):
             return False