No. Settings in the three dots at the top of Copilot Chat open the same global settings as the menu at the bottom of the screen. Any change you make there applies to all your chats. The three dots only show this option on a new, empty chat.
More in Microsoft Copilot
Can I tell Copilot how to name my chats and Pages?
Chats yes, Pages no. Put a naming rule in your custom instructions or save it as a memory, and new chats then follow it in your chat history. Pages are different: Copilot names them in a premade way before it reads your instructions, so rename each page yourself into whatever structure you want.
What is the Library in Copilot?
The Library, in Copilot's left-hand bar, is where Copilot stores what it creates: your Pages, canvases, generated images and other creations. It is storage, not a working feature, and it does not run prompts. Pages are saved there automatically, and a page survives even if you delete the chat it came from.
Can my colleagues or IT see my Microsoft Copilot chats?
Your colleagues cannot see your Copilot chats. Microsoft 365 Copilot only works inside your own permissions, so nobody can ask it about your mail or files. Your organisation can still keep and search Copilot conversations for compliance and audit through Microsoft Purview, the same way it can with email, so treat work chats as work records.