Change title of untitled tab in Visual Studio Code

It is still (Q1 2020) not possible, but the next VSCode 1.42 will name its Untitled editors differently.

Untitled editors in VS Code are text buffers that have not yet been saved to disk.
You can leave them open for as long as you like and all text content is stored and restored between restarts.

Untitled editors were given generic names such as Untitled-1 and counting upwards.
In this release, untitled editors will use the content of the first line of the document for the editor title, and include the generic name as part of the description:

Note: If the first line is empty or does not contain any words, the title will fall back to Untitled_* as before.

So while you cannot set the title yourself (still readonly fileName), technically... changing the first line of that file would be enough to change the title of said "Untitled" editor.

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With VSCode 1.43 (Q1 2020), a new setting workbench.editor.untitled.labelFormat allows to control whether untitled editors should use the contents as title or not.
Possible values are content or name.
Configure 'workbench.editor.untitled.labelFormat': 'name' to get the previous behavior back where untitled editors would have a short title, such as Untitled-1.

It's not possible - if you check out the source code for the API definition in vscode.d.ts, you'll see that fileName is declared as readonly:

export interface TextDocument {
    // ...
    readonly fileName: string;
    // ...
}

Unfortunately, it seems that the readonly attribute isn't reflected in the API docs on the website.