$$ x \cos \alpha + y \sin \alpha -p = 0$$ represents a straight line in polar form (or even taken in any other form),
$$ (x \cos \alpha + y \sin \alpha -p )^3 = 0$$ represents 3 straight lines repeated, but why does not, $ (n\in \mathbb Z) $
$$ ( x \cos \alpha + y \sin \alpha -p)^{2\,n} = 0 $$ at all plot for even powers ? I used Mathematica but other CAS could be written the same way in this respect. Why does it not represent evenly repeated straight lines?
There are two distinct issues here: what the equation represents and what plots (on particular software).
For any positive integer $n$, $f(x,y)^n = 0$ has exactly the same solutions as $f(x,y) = 0$.
On typical software programs that use numerical methods, implicit plots depend on detecting sign changes, so they will often miss local minima of $0$. See e.g. this recent question