What exactly does it mean that the key $e \in \mathcal K$ uniquely determines $E_e$ ?
Does it mean that for each $e \in \mathcal K$ there exist only one function $E_e$ corresponding to $e$ ?
Does this imply that two different keys can correspond to the same function ? I mean, this doesn't violate that there only correspond one function to each key ?
It means that the function $E_e$ as a whole (as a bijective function from the message space $\mathcal{M}$ to the ciphertext space $\mathcal{C}$) is determined by the key $e$ alone. This implies that for every $e \in \mathcal{K}$ and every $m \in \mathcal{M}$ there is a uniquely determined $E_e(m) \in \mathcal{C}$. So yes to your first question.
This doesn't exclude the possibility that two different $e,e'$ can determine the same functions $E_e = E_{e'}$ (although this is unusual to happen in practice). This could happen, and is not in violation with the previous fact.