I don't remember this being the case. I don't think it was possible to change either a file's type or creator code without a tool such as ResEdit that wasn't included in the OS.
BTW, I think you're referring to the 'creator' code, not the 'type' code. The 'creator' code determined which application would open a file when it was opened. And it was per-file; I don't remember any mention of a system-wide 'default application' or anything like that.
This was actually rather nice in practice. It meant a JPEG image created by e.g., GraphicsConverter would be opened by GraphicsConverter when double-clicked; whereas one saved by a web browser would be opened in the web browser. But either could be dragged into either application in order to open the web-browser-saved image with GraphicsConverter.