diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 51ca9dbd763b4..6b7aca1b6c4ee 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -10420,12 +10420,14 @@ def _count_level(self, level: Level, axis: int = 0, numeric_only: bool = False):
         else:
             mask = index_mask.reshape(-1, 1) & values_mask
 
-        if isinstance(level, str):
-            level = count_axis._get_level_number(level)
+        if isinstance(level, int):
+            level_number = level
+        else:
+            level_number = count_axis._get_level_number(level)
 
-        level_name = count_axis._names[level]
-        level_index = count_axis.levels[level]._rename(name=level_name)
-        level_codes = ensure_platform_int(count_axis.codes[level])
+        level_name = count_axis._names[level_number]
+        level_index = count_axis.levels[level_number]._rename(name=level_name)
+        level_codes = ensure_platform_int(count_axis.codes[level_number])
         counts = lib.count_level_2d(mask, level_codes, len(level_index), axis=axis)
 
         if axis == 1:
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 752ce28c58f55..29df930c5aaf3 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -287,7 +287,7 @@ class MultiIndex(Index):
 
     # initialize to zero-length tuples to make everything work
     _typ = "multiindex"
-    _names = FrozenList()
+    _names: list[Hashable | None] = []
     _levels = FrozenList()
     _codes = FrozenList()
     _comparables = ["names"]
@@ -326,9 +326,7 @@ def __new__(
         result._set_levels(levels, copy=copy, validate=False)
         result._set_codes(codes, copy=copy, validate=False)
 
-        # Incompatible types in assignment (expression has type "List[None]",
-        # variable has type "FrozenList")  [assignment]
-        result._names = [None] * len(levels)  # type: ignore[assignment]
+        result._names = [None] * len(levels)
         if names is not None:
             # handles name validation
             result._set_names(names)
@@ -1476,8 +1474,7 @@ def _set_names(self, names, *, level=None, validate: bool = True):
                     raise TypeError(
                         f"{type(self).__name__}.name must be a hashable type"
                     )
-            # error: Cannot determine type of '__setitem__'
-            self._names[lev] = name  # type: ignore[has-type]
+            self._names[lev] = name
 
         # If .levels has been accessed, the names in our cache will be stale.
         self._reset_cache()