Complete deprecation cycle for Dataset.update return value (#10717)

* Complete deprecation cycle for Dataset.update return value

- Dataset.update() now returns None instead of self
- Removes deprecation that was started in v0.17 and scheduled for removal in v0.21
- Updates documentation to remove return value information
- Aligns behavior with standard dict.update() which returns None

Closes #10167

* Update doc/whats-new.rst

Co-authored-by: Stephan Hoyer <shoyer@google.com>

---------

Co-authored-by: Stephan Hoyer <shoyer@google.com>

Author: max-sixty

doc/whats-new.rst

@@ -17,6 +17,10 @@ New Features
 Breaking changes
 ~~~~~~~~~~~~~~~~
 
+- :py:meth:`Dataset.update` now returns ``None``, instead of the updated dataset. This
+  completes the deprecation cycle started in version 0.17. The method still updates the
+  dataset in-place. (:issue:`10167`)
+  By `Maximilian Roos <https://github.com/max-sixty>`_.
 
 Deprecations
 ~~~~~~~~~~~~

xarray/core/dataset.py

@@ -5592,7 +5592,7 @@ def unstack(
                 result = result._unstack_once(d, stacked_indexes[d], fill_value, sparse)
         return result
 
-    def update(self, other: CoercibleMapping) -> Self:
+    def update(self, other: CoercibleMapping) -> None:
         """Update this dataset's variables with those from another dataset.
 
         Just like :py:meth:`dict.update` this is a in-place operation.
@@ -5609,14 +5609,6 @@ def update(self, other: CoercibleMapping) -> Self:
             - mapping {var name: (dimension name, array-like)}
             - mapping {var name: (tuple of dimension names, array-like)}
 
-        Returns
-        -------
-        updated : Dataset
-            Updated dataset. Note that since the update is in-place this is the input
-            dataset.
-
-            It is deprecated since version 0.17 and scheduled to be removed in 0.21.
-
         Raises
         ------
         ValueError
@@ -5629,7 +5621,7 @@ def update(self, other: CoercibleMapping) -> Self:
         Dataset.merge
         """
         merge_result = dataset_update_method(self, other)
-        return self._replace(inplace=True, **merge_result._asdict())
+        self._replace(inplace=True, **merge_result._asdict())
 
     def merge(
         self,