I have seen that there is a similar question here but none that really answers the question. I understand that if I choose the encryption exponent $e$ not coprime with $\varphi(n)$ then there is not a unique way to decrypt a message.

What I am wondering is what is the mathematical reason behind this? It seems to me that since $m^{(k \varphi(n)+1)} = m \bmod N$ and $d$ is defined as $(k\varphi(n)+1)/e$ then $d\cdot e$ is always going to be $k\varphi(n) +1$. What am I missing?