What are you hoping to get out of a postdoc? Typically postdocs offer a chance to enhance your research credentials and capabilities before moving on to a more permanent position either in academia or a serious research institution. If you do not have a PhD and wish to follow this career path, the best way to enhance your credentials and capabilities would be to pursue a PhD. A very capable PhD student can certainly do work of equal quality to postdoctoral researchers while also gaining a degree in the process. The only drawbacks to a PhD position compared to a postdoc seem to be somewhat lower pay and receiving slightly less individual credit for your work (in some fields anyway).
To more directly answer your question: Given enough funding flexibility, it's not unheard of for a professor to fill an advertised postdoctoral position with a very capable grad student. It would be very uncommon for a non-PhD, non-grad student to hold such a position.
As an aside, by definition you cannot hold a postdoctoral position without having a doctorate. The term literally means "after doctorate" and therefore requiress first gaining a doctorate. Any position you hold before obtaining a doctorate is by definition a pre-doctoral position.