Pages automatically showing extension when opening new documents

You should skim over the existing posts to see if your issue has already been addressed as it has in:

.pages and .numbers filename extensions c… - Apple Community

Pages has nothing to do with showing or hiding Pages document extensions. If the document extension was hidden in the Finder Get Info panel, then Pages will open the document without showing the extension in the title bar. There is no Finder setting that hide all file extensions.

Current versions of Pages always save a document with the extension showing. There is no override in Pages to not do that, as it is part of the application design.

Those transitioning from Windows and its Explorer are accustomed to documents without extensions, but macOS and its Finder have nothing to do with Microsoft's design decisions. Adapt.

One of the things that I have noticed with Pages v14.4 on Sequoia v15.5 is that if the last document save location was in the Documents folder, and one creates a new Pages document from a blank template, and then looks for File menu > Save… to initially save the document, one only encounters Save and no Save dialog is presented. That document is saved as Untitled with no extension. That is a significant departure from past Pages version behavior.

If the last saved location was on the Desktop, then one will encounter a File menu > Save… menu item upon first saving the document, and the resulting Save dialog shows Untitled.pages ready for one to give it a proper filename. There is no setting in Pages or the Finder to automatically hide file extensions.

The last saved document location should not modify the File menu > Save… menu item for first document saves.

Many years ago, Pages used to offer an option on the Save dialog allowing the user control over the ability to hide or show the extension. No longer.

In the Finder, one can select one or more Pages documents with extensions shown and press option+cmd+i to open the Get Info panel. By clicking on the Hide extension option, the selected files' extensions will be hidden in the Finder.

My own preference is to use the Mac as a Mac and not emulate Microsoft Windows Explorer by hiding extensions.