Will the nth Kissing Number always be the number of vertices of an uniform (or regular) n-polytope? For example, the 3rd Kissing Number, 12, is the number of vertices of an icosaedron; the 4th one, 24, is the number of vertices of a 16-cell truncation (and many others); the 8th one, 240, is the number of vertices of the $4_{21}$ 8-polytope (of the $E_8$ group), etc.
Is it possible to make a generalization of this phenomen or is it actually nothing but a coincidence?
 A: To expand on my comment regarding the regular polytope version of this question, the answer is "no".
The kissing numbers are unknown, although crude bounds to them are known. From this page one may read that the kissing number in dimension $n=5$ is between $40$ and $44$. 
The regular polytopes are known. In dimension $n \ge 5$, there are only three of them: the simplex with $n+1$ vertices, the cube with $2^n$ vertices, and the "orthohedron" (generalizing the octahedron) with $2n$ vertices. In dimension $n=5$, these three numbers are $5+1=6$, $2^5=32$, and $2 \cdot 5 = 10$. None of these three numbers are between $40$ and $44$. 
One might wonder if something like this could "eventually" be true, i.e. be true for sufficiently large $n$. But the answer still appears to be "no", although I'm not sure whether this can yet be proved rigorously (as hinted in the comment of @MJD). Here's why.
On the wikipedia page cited above, you will see that the kissing numbers are believed to grow exponentially, which rules out $n+1$ and $2n$. You will also see from the table and the graph reproduced on that wikipedia page that although the base of the exponential is unknown, it appears to be strictly less than $2$, which rules out $2^n$. 
Since there is as yet no proof of any of these asymptotic estimates (as far as I can tell in my quick and dirty reading of wikipedia), this all remains unknown in any rigorous sense. 
