diff --git a/xarray/core/concat.py b/xarray/core/concat.py
index 40ae0b12bc9..b59cee4491a 100644
--- a/xarray/core/concat.py
+++ b/xarray/core/concat.py
@@ -318,13 +318,17 @@ def determine_dims(datasets, result_coord_names):
     both_data_and_coords = result_coord_names & noncoord_names
     if both_data_and_coords:
         raise ValueError(
-            "%r is a coordinate in some datasets and not in others."
-            % list(both_data_and_coords)[0]  # preserve previous behaviour
+            "%r is a coordinate in some datasets but not others."
+            % list(both_data_and_coords)[0]  # preserve format of error message
         )
     dim_names, result_coords = determine_dims(datasets, result_coord_names)
     # we don't want the concat dimension in the result dataset yet
     result_coords.pop(dim, None)
 
+    # case where concat dimension is a coordinate but not a dimension
+    if dim in result_coord_names and dim not in dim_names:
+        datasets = [ds.expand_dims(dim) for ds in datasets]
+
     # determine which variables to concatentate
     concat_over, equals = _calc_concat_over(
         datasets, dim, dim_names, data_vars, coords, compat
@@ -335,6 +339,10 @@ def determine_dims(datasets, result_coord_names):
     if variables_to_merge:
         to_merge = []
         for ds in datasets:
+            if variables_to_merge - set(ds.variables):
+                raise ValueError(
+                    "Encountered unexpected variables %r" % list(variables_to_merge)[0]
+                )
             to_merge.append(ds.reset_coords()[list(variables_to_merge)])
         # TODO: Provide equals as an argument and thread that down to merge.unique_variable
         result_vars = merge_variables(
@@ -397,6 +405,11 @@ def ensure_common_dims(vars):
     result = result.set_coords(result_coord_names)
     result.encoding = result_encoding
 
+    # TODO: avoid this
+    unlabeled_dims = dim_names - result_coord_names
+    result = result.drop(unlabeled_dims, errors="ignore")
+
+    # need to pass test when provided dim is a DataArray
     if coord is not None:
         # add concat dimension last to ensure that its in the final Dataset
         result[coord.name] = coord
diff --git a/xarray/tests/test_concat.py b/xarray/tests/test_concat.py
index 78373577efd..c3b48c4b2ca 100644
--- a/xarray/tests/test_concat.py
+++ b/xarray/tests/test_concat.py
@@ -5,8 +5,7 @@
 import pytest
 
 from xarray import DataArray, Dataset, Variable, concat
-from xarray.core import dtypes
-
+from xarray.core import dtypes, merge
 from . import (
     InaccessibleArray,
     assert_array_equal,
@@ -36,11 +35,15 @@ def test_concat_compat():
         coords={"x": [0, 1], "y": [1], "z": [-1, -2], "q": [0]},
     )
 
-    result = concat([ds1, ds2], dim="y", compat="equals")
+    result = concat([ds1, ds2], dim="y", data_vars="minimal", compat="broadcast_equals")
+    assert_equal(ds2.no_x_y, result.no_x_y.transpose())
 
-    for var in ["has_x", "no_x_y", "const1"]:
+    for var in ["has_x", "no_x_y"]:
         assert "y" not in result[var]
 
+    with raises_regex(ValueError, "'q' is not a dimension in all datasets"):
+        concat([ds1, ds2], dim="q", data_vars="all", compat="broadcast_equals")
+
 
 class TestConcatDataset:
     @pytest.fixture
@@ -116,7 +119,7 @@ def test_concat_coords(self):
             actual = concat(objs, dim="x", coords=coords)
             assert_identical(expected, actual)
         for coords in ["minimal", []]:
-            with raises_regex(ValueError, "not equal across"):
+            with raises_regex(merge.MergeError, "conflicting values"):
                 concat(objs, dim="x", coords=coords)
 
     def test_concat_constant_index(self):
@@ -127,8 +130,10 @@ def test_concat_constant_index(self):
         for mode in ["different", "all", ["foo"]]:
             actual = concat([ds1, ds2], "y", data_vars=mode)
             assert_identical(expected, actual)
-        with raises_regex(ValueError, "not equal across datasets"):
-            concat([ds1, ds2], "y", data_vars="minimal")
+        with raises_regex(merge.MergeError, "conflicting values"):
+            # previously dim="y", and raised error which makes no sense.
+            # "foo" has dimension "y" so minimal should concatenate it?
+            concat([ds1, ds2], "new_dim", data_vars="minimal")
 
     def test_concat_size0(self):
         data = create_test_data()
@@ -372,10 +377,10 @@ def setUp(self):
 
     def test_concat_sensible_compat_errors(self):
 
-        with raises_regex(ValueError, "'only_x' is not equal across datasets."):
+        with raises_regex(merge.MergeError, "conflicting values"):
             concat(self.dsets, data_vars="sensible", dim="y")
 
-        with raises_regex(ValueError, "'coord1' is not equal across datasets."):
+        with raises_regex(merge.MergeError, "conflicting values"):
             concat(self.dsets, coords="sensible", dim="y")
 
     @pytest.mark.parametrize("concat_dim", ["y", "new_dim"])
@@ -431,7 +436,9 @@ def test_compat_override(self, data_vars, coords):
 
     def test_compat_override_different_error(self):
         with raises_regex(ValueError, "Cannot specify both .*='different'"):
-            concat(self.dsets, data_vars="different", compat="override")
+            concat(
+                self.dsets, dim="concat_dim", data_vars="different", compat="override"
+            )
 
 
 class TestConcatDataArray: