I think that depends what you consider a bug. I thinj Mario not being present makes sense as he's dead. The princess was saved even though Mario died, so I think it's reasonable to say the win condition is met. What I think is definitely a bug is that you can hear the death sound way late (a 2nd time) when you do this (jump and hit the axe at the same time bowser touches you).

It's a testament to the quality of the programming and testing that they show Mario on the screen only if he's still alive. It would be easier to just show him without conditions.