Fix registering a blueprint twice with differing names

Previously the blueprint recorded aspects (before request, after
request etc) would only be added to the app if it was the first
registration of the blueprint instance. However only the record-once
aspects (app-before requests, app-after request) should be added once
on registration of the instance, whereas everything else should be
added on every unique name registration. This ensures that these
trigger under the new name as well as the old.

CHANGES.rst

@@ -12,6 +12,7 @@ Unreleased
     decorators to accept functions with no arguments. :issue:`4098`
 -   Support View and MethodView instances with async handlers. :issue:`4112`
 -   Enhance typing of ``app.errorhandler`` decorator. :issue:`4095`
+-   Fix registering a blueprint twice with differing names. :issue:`4124`
 
 
 Version 2.0.1

src/flask/blueprints.py

@@ -293,7 +293,6 @@ class Blueprint(Scaffold):
             Registering the same blueprint with the same name multiple
             times is deprecated and will become an error in Flask 2.1.
         """
-        first_registration = not any(bp is self for bp in app.blueprints.values())
         name_prefix = options.get("name_prefix", "")
         self_name = options.get("name", self.name)
         name = f"{name_prefix}.{self_name}".lstrip(".")
@@ -318,9 +317,12 @@ class Blueprint(Scaffold):
                     stacklevel=4,
                 )
 
+        first_bp_registration = not any(bp is self for bp in app.blueprints.values())
+        first_name_registration = name not in app.blueprints
+
         app.blueprints[name] = self
         self._got_registered_once = True
-        state = self.make_setup_state(app, options, first_registration)
+        state = self.make_setup_state(app, options, first_bp_registration)
 
         if self.has_static_folder:
             state.add_url_rule(
@@ -330,7 +332,7 @@ class Blueprint(Scaffold):
             )
 
         # Merge blueprint data into parent.
-        if first_registration:
+        if first_bp_registration or first_name_registration:
 
             def extend(bp_dict, parent_dict):
                 for key, values in bp_dict.items():

tests/test_blueprints.py

@@ -899,6 +899,14 @@ def test_blueprint_renaming(app, client) -> None:
     def index():
         return flask.request.endpoint
 
+    @bp.get("/error")
+    def error():
+        flask.abort(403)
+
+    @bp.errorhandler(403)
+    def forbidden(_: Exception):
+        return "Error", 403
+
     @bp2.get("/")
     def index2():
         return flask.request.endpoint
@@ -911,3 +919,5 @@ def test_blueprint_renaming(app, client) -> None:
     assert client.get("/b/").data == b"alt.index"
     assert client.get("/a/a/").data == b"bp.sub.index2"
     assert client.get("/b/a/").data == b"alt.sub.index2"
+    assert client.get("/a/error").data == b"Error"
+    assert client.get("/b/error").data == b"Error"