Fixes Agent ID not displaying new name on examine proc (#22465)

See title; set name action was only updating one of the card's "name"
variables, not both.

code/game/objects/items/weapons/cards_ids_syndicate.dm

@@ -179,8 +179,10 @@
 			if(!isnull(newName))
 				if(newName == "")
 					newName = initial(name)
+					registered_name = initial(registered_name)
 				else
 					name = newName
+					registered_name = newName
 				to_chat(user, SPAN_NOTICE("Name has been set to '[name]'."))
 		if("setAge")
 			var/newAge = params["age"]

html/changelogs/Bat-AgentIDName.yml

@@ -0,0 +1,4 @@
+author: Batrachophrenoboocosmomachia
+delete-after: True
+changes:
+  - bugfix: "Agent ID name changes now correctly set both internal name variables."