diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 3740a5a44f1..a23883338b0 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -21,6 +21,10 @@ v2023.07.1 (unreleased)
 
 New Features
 ~~~~~~~~~~~~
+- Allow creating Xarray objects where a multidimensional variable shares its name
+  with a dimension. Examples include output from finite volume models like FVCOM.
+  (:issue:`2233`, :pull:`7989`)
+  By `Deepak Cherian <https://github.com/dcherian>`_ and `Benoit Bovy <https://github.com/benbovy>`_.
 
 
 Breaking changes
diff --git a/xarray/core/indexes.py b/xarray/core/indexes.py
index 9ee9bc374d4..7159178aa3c 100644
--- a/xarray/core/indexes.py
+++ b/xarray/core/indexes.py
@@ -1362,7 +1362,7 @@ def default_indexes(
     coord_names = set(coords)
 
     for name, var in coords.items():
-        if name in dims:
+        if name in dims and var.ndim == 1:
             index, index_vars = create_default_index_implicit(var, coords)
             if set(index_vars) <= coord_names:
                 indexes.update({k: index for k in index_vars})
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 9271d0c4dbd..f2b05bb441f 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -158,14 +158,8 @@ def as_variable(obj, name=None) -> Variable | IndexVariable:
             f"explicit list of dimensions: {obj!r}"
         )
 
-    if name is not None and name in obj.dims:
-        # convert the Variable into an Index
-        if obj.ndim != 1:
-            raise MissingDimensionsError(
-                f"{name!r} has more than 1-dimension and the same name as one of its "
-                f"dimensions {obj.dims!r}. xarray disallows such variables because they "
-                "conflict with the coordinates used to label dimensions."
-            )
+    if name is not None and name in obj.dims and obj.ndim == 1:
+        # automatically convert the Variable into an Index
         obj = obj.to_index_variable()
 
     return obj
diff --git a/xarray/testing.py b/xarray/testing.py
index b6a88135ee1..80c73430312 100644
--- a/xarray/testing.py
+++ b/xarray/testing.py
@@ -364,14 +364,13 @@ def _assert_dataset_invariants(ds: Dataset, check_default_indexes: bool):
     assert all(
         ds._dims[k] == v.sizes[k] for v in ds._variables.values() for k in v.sizes
     ), (ds._dims, {k: v.sizes for k, v in ds._variables.items()})
-    assert all(
-        isinstance(v, IndexVariable)
-        for (k, v) in ds._variables.items()
-        if v.dims == (k,)
-    ), {k: type(v) for k, v in ds._variables.items() if v.dims == (k,)}
-    assert all(v.dims == (k,) for (k, v) in ds._variables.items() if k in ds._dims), {
-        k: v.dims for k, v in ds._variables.items() if k in ds._dims
-    }
+
+    if check_default_indexes:
+        assert all(
+            isinstance(v, IndexVariable)
+            for (k, v) in ds._variables.items()
+            if v.dims == (k,)
+        ), {k: type(v) for k, v in ds._variables.items() if v.dims == (k,)}
 
     if ds._indexes is not None:
         _assert_indexes_invariants_checks(
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index 01c26ad6104..f7f91d0e134 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -35,6 +35,7 @@
 from xarray.core.indexes import Index, PandasIndex
 from xarray.core.pycompat import array_type, integer_types
 from xarray.core.utils import is_scalar
+from xarray.testing import _assert_internal_invariants
 from xarray.tests import (
     DuckArrayWrapper,
     InaccessibleArray,
@@ -467,13 +468,16 @@ def test_constructor(self) -> None:
 
         with pytest.raises(ValueError, match=r"conflicting sizes"):
             Dataset({"a": x1, "b": x2})
-        with pytest.raises(ValueError, match=r"disallows such variables"):
-            Dataset({"a": x1, "x": z})
         with pytest.raises(TypeError, match=r"tuple of form"):
             Dataset({"x": (1, 2, 3, 4, 5, 6, 7)})
         with pytest.raises(ValueError, match=r"already exists as a scalar"):
             Dataset({"x": 0, "y": ("x", [1, 2, 3])})
 
+        # nD coordinate variable "x" sharing name with dimension
+        actual = Dataset({"a": x1, "x": z})
+        assert "x" not in actual.xindexes
+        _assert_internal_invariants(actual, check_default_indexes=True)
+
         # verify handling of DataArrays
         expected = Dataset({"x": x1, "z": z})
         actual = Dataset({"z": expected["z"]})
diff --git a/xarray/tests/test_variable.py b/xarray/tests/test_variable.py
index a6dffb82660..9b70dcb5464 100644
--- a/xarray/tests/test_variable.py
+++ b/xarray/tests/test_variable.py
@@ -1247,8 +1247,10 @@ def test_as_variable(self):
         expected = Variable(("x", "y"), data)
         with pytest.raises(ValueError, match=r"without explicit dimension names"):
             as_variable(data, name="x")
-        with pytest.raises(ValueError, match=r"has more than 1-dimension"):
-            as_variable(expected, name="x")
+
+        # name of nD variable matches dimension name
+        actual = as_variable(expected, name="x")
+        assert_identical(expected, actual)
 
         # test datetime, timedelta conversion
         dt = np.array([datetime(1999, 1, 1) + timedelta(days=x) for x in range(10)])