Although there are similarities in the 2 pieces of music, they are technically unrelated. This type of thing happens more than you would think, in all kinds of music. When discussing music, it helps to be as objective as possible. So here's what i see

Both pieces are in a "minor key/mode." This just means that there are conventions at play that will affect the listener psychologically by their associations.
Punch out - a minor
SK Theme - c minor

Both pieces feature a chord progression of VI - VII - i - VII - VI in their respective minor keys (props to your ears for hearing this)
Punch out - this is the main progression of the piece (in this case, the pattern is: i - VII - VI - VII)
SK Theme - this progression appears only for a moment at 0:38, and is modified on its repeat.

Both pieces have a steady bass line, riding on the divisions of the beat but staying on the same note. The tempos are close also.
The Punch Out theme's bass line is way more basic, staying on its pitch. The SK Theme's bass line moves around the notes of the chord, but although there is movment, the line is fairly static as well.

Although these similarities exist, it isn't enough to call it the "same" music. For the differences are many: most importantly, the melody (which is what copyright usually goes by). If you sing just the melody to both consecutively, you will be hard pressed to find anything similar. The orchestration is totally different also (which is obvious enough). Your ears picked up most if not all of this, I'm sure. At the end of the day though, we just chalk it up to "in-game music" having some standardized methods that work quite well! (in both cases!) :D

cheers!