The Gemoro (Menochos 33a) says עשאה כמין נגר פסולה
if you placed the Mezuzah like a bolt/peg it is Pasul.
The Machlokes between Rashi & R.Tam is what type of
a bolt the Gemoro is referring to. (vertical or horizontal)
The Machlokes is not which position one should place
the Mezuzah but which position not to place the
Mezuzah so that it is not כמין נגר.
All agree as long it is not positioned like a peg it is kosher.
Positioning the Mezuza at an angle is not a compromise
(doing it halfway like both) but is the proper Lekatchila
way to place it,(according to both) being a peg is never
placed at an angle.
Therefore when placed at an angle there is no need to
place two Mezuzos. It is positioned according to both
Rashi & R.Tam in the proper position.
You are יוצא ידי שניהם by placing it at an angle
This solution won't work by Tefillin.
Placing the Parshiyos in the Tefillin the Halacha dictates
how to place the Parshiyos and not how not to place
it. Therefore by placing it at an angle both Rashi &
R. Tam will agree that it is Pasul. אמבצי
The Gaon who holds to place the Mezuzah vertically
(not angle) also holds it's not necessary to put on
R.Tam's Teffilin. In both cases since the Halacha is like
Rashi there is no need to be יוצא ידי שניהם