A very common bronze age interpretation of Superman's powers are that most of them stem from a "bio-energy matrix", a field of energy surrounding his body that is able to nullify or amplify the effect of other energy fields it comes in contact with (this field is powered by solar energy, stored within his body. That's why Batman blocks out the sun in Dark Knight Returns to drain Supes of his powers temporarily). For an example of this, he flies by neutralizing gravity.
This field is controlled by him subconsciously, it's like a reflex. This field can be extended to other objects as well, which is why he doesn't compromise the structural integrity of buildings when he catches them, because he extends his gravity nullification to surround the building.
He's also been shown in some cases to be able to remove this field temporarily at will, which would be how he shaves with regular tools.
Another common, but very unlikely, assertion is that he is unable to remove the field at will, and thus has to shave with either 1. a kryptonite coated razor or 2. a magically imbued razor, as those are the only external influences known to be able to either cancel out or ignore the bioenergy matrix.
That, or his bathroom light is red sun (which neutralizes the yellow sun energy stored in his body).